Rosa's DopplerBench

Percentage Accurate: 73.4% → 98.1%

Time: 11.4s

Alternatives: 11

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.08e+159)
   (/ v (- t1))
   (if (<= t1 1.7e+128)
     (* v (/ t1 (* (+ t1 u) (- (- u) t1))))
     (- (/ v (fma u 2.0 t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = v / -t1;
	} else if (t1 <= 1.7e+128) {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	} else {
		tmp = -(v / fma(u, 2.0, t1));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.08e+159)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 1.7e+128)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))));
	else
		tmp = Float64(-Float64(v / fma(u, 2.0, t1)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.08e+159], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 1.7e+128], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.07999999999999991e159

   1. Initial program 47.1%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.07999999999999991e159 < t1 < 1.6999999999999999e128

   1. Initial program 80.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

   if 1.6999999999999999e128 < t1

   1. Initial program 45.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. clear-num N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

   6. Applied rewrites 94.7%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 93.0

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

   9. Applied rewrites 93.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (fma u 2.0 t1)))))
   (if (<= t1 -1.8e-117)
     t_1
     (if (<= t1 4.8e-12) (* v (/ (/ t1 (- u)) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / fma(u, 2.0, t1));
	double tmp;
	if (t1 <= -1.8e-117) {
		tmp = t_1;
	} else if (t1 <= 4.8e-12) {
		tmp = v * ((t1 / -u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / fma(u, 2.0, t1)))
	tmp = 0.0
	if (t1 <= -1.8e-117)
		tmp = t_1;
	elseif (t1 <= 4.8e-12)
		tmp = Float64(v * Float64(Float64(t1 / Float64(-u)) / u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -1.8e-117], t$95$1, If[LessEqual[t1, 4.8e-12], N[(v * N[(N[(t1 / (-u)), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.8e-117 or 4.79999999999999974e-12 < t1

   1. Initial program 60.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. clear-num N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 81.7

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

   9. Applied rewrites 81.7%

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

   if -1.8e-117 < t1 < 4.79999999999999974e-12

   1. Initial program 82.4%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 80.9

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

   7. Applied rewrites 80.9%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-neg.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 84.3

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

   9. Applied rewrites 84.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{v}{t1 + u} \cdot \left(-t1\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.2e+167)
   (/ (* (/ v (+ t1 u)) (- t1)) u)
   (/ (- v) (fma u (+ 2.0 (/ u t1)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.2e+167) {
		tmp = ((v / (t1 + u)) * -t1) / u;
	} else {
		tmp = -v / fma(u, (2.0 + (u / t1)), t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.2e+167)
		tmp = Float64(Float64(Float64(v / Float64(t1 + u)) * Float64(-t1)) / u);
	else
		tmp = Float64(Float64(-v) / fma(u, Float64(2.0 + Float64(u / t1)), t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.2e+167], N[(N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * (-t1)), $MachinePrecision] / u), $MachinePrecision], N[((-v) / N[(u * N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] + t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.2000000000000004e167

   1. Initial program 75.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      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. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -5.2000000000000004e167 < u

   1. Initial program 67.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. clear-num N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

   6. Applied rewrites 96.7%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 96.7

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

   9. Applied rewrites 96.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{v}{t1 + u} \cdot \left(-t1\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (fma u 2.0 t1)))))
   (if (<= t1 -1.8e-117)
     t_1
     (if (<= t1 4.8e-12) (* v (/ t1 (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / fma(u, 2.0, t1));
	double tmp;
	if (t1 <= -1.8e-117) {
		tmp = t_1;
	} else if (t1 <= 4.8e-12) {
		tmp = v * (t1 / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / fma(u, 2.0, t1)))
	tmp = 0.0
	if (t1 <= -1.8e-117)
		tmp = t_1;
	elseif (t1 <= 4.8e-12)
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -1.8e-117], t$95$1, If[LessEqual[t1, 4.8e-12], N[(v * N[(t1 / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.8e-117 or 4.79999999999999974e-12 < t1

   1. Initial program 60.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. clear-num N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 81.7

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

   9. Applied rewrites 81.7%

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

   if -1.8e-117 < t1 < 4.79999999999999974e-12

   1. Initial program 82.4%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 81.5

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

   7. Applied rewrites 81.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (fma u 2.0 t1)))))
   (if (<= t1 -2e-42) t_1 (if (<= t1 4.8e-12) (* t1 (/ v (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / fma(u, 2.0, t1));
	double tmp;
	if (t1 <= -2e-42) {
		tmp = t_1;
	} else if (t1 <= 4.8e-12) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / fma(u, 2.0, t1)))
	tmp = 0.0
	if (t1 <= -2e-42)
		tmp = t_1;
	elseif (t1 <= 4.8e-12)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -2e-42], t$95$1, If[LessEqual[t1, 4.8e-12], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.00000000000000008e-42 or 4.79999999999999974e-12 < t1

   1. Initial program 59.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. clear-num N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 86.0

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

   9. Applied rewrites 86.0%

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

   if -2.00000000000000008e-42 < t1 < 4.79999999999999974e-12

   1. Initial program 80.9%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 72.8

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

   5. Applied rewrites 72.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{-42}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 8: 58.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{-u}\\ \mathbf{if}\;u \leq -1.5 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+213}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- u))))
   (if (<= u -1.5e+224) t_1 (if (<= u 3.1e+213) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / -u;
	double tmp;
	if (u <= -1.5e+224) {
		tmp = t_1;
	} else if (u <= 3.1e+213) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / -u
    if (u <= (-1.5d+224)) then
        tmp = t_1
    else if (u <= 3.1d+213) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / -u;
	double tmp;
	if (u <= -1.5e+224) {
		tmp = t_1;
	} else if (u <= 3.1e+213) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / -u
	tmp = 0
	if u <= -1.5e+224:
		tmp = t_1
	elif u <= 3.1e+213:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(-u))
	tmp = 0.0
	if (u <= -1.5e+224)
		tmp = t_1;
	elseif (u <= 3.1e+213)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / -u;
	tmp = 0.0;
	if (u <= -1.5e+224)
		tmp = t_1;
	elseif (u <= 3.1e+213)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / (-u)), $MachinePrecision]}, If[LessEqual[u, -1.5e+224], t$95$1, If[LessEqual[u, 3.1e+213], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.5000000000000001e224 or 3.09999999999999991e213 < u

   1. Initial program 78.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 58.4

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

   10. Applied rewrites 58.4%

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

   if -1.5000000000000001e224 < u < 3.09999999999999991e213

   1. Initial program 66.1%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 65.6

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

   5. Applied rewrites 65.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{+224}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+213}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-+.f64 N/A

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

   2. div-inv N/A

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

   3. div-inv N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. clear-num N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. associate-/r/ N/A

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

   14. associate-*l/ N/A

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

   15. *-lft-identity N/A

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

6. Applied rewrites 94.8%

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

7. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 66.9

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

9. Applied rewrites 66.9%

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

10. Final simplification 66.9%

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

Alternative 10: 61.7% accurate, 1.8× speedup

Math

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

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(v / Float64(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 68.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   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. lower-neg.f64 97.7

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in t1 around inf

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

   1. lower-/.f64 66.6

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

7. Applied rewrites 66.6%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. lift-+.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. times-frac N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. neg-mul-1 N/A

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

   10. *-inverses N/A

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

   11. lift-neg.f64 N/A

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

   12. div-inv N/A

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

   13. lower-/.f64 66.6

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

9. Applied rewrites 66.6%

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

10. Final simplification 66.6%

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

Alternative 11: 17.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in t1 around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 50.3

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

7. Applied rewrites 50.3%

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

8. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 20.2

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

10. Applied rewrites 20.2%

    \[\leadsto \color{blue}{\frac{v}{-u}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))