Rosa's DopplerBench

Percentage Accurate: 73.0% → 98.1%

Time: 10.8s

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: 73.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-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 98.5

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u \cdot -2, v\right)}{-t1}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)))
   (if (<= t1 -9.2e+122)
     (/ (fma (/ v t1) (* u -2.0) v) (- t1))
     (if (<= t1 2.1e+121) (* v (/ t1 (* (+ t1 u) t_1))) (/ v t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (t1 <= -9.2e+122) {
		tmp = fma((v / t1), (u * -2.0), v) / -t1;
	} else if (t1 <= 2.1e+121) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	tmp = 0.0
	if (t1 <= -9.2e+122)
		tmp = Float64(fma(Float64(v / t1), Float64(u * -2.0), v) / Float64(-t1));
	elseif (t1 <= 2.1e+121)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, If[LessEqual[t1, -9.2e+122], N[(N[(N[(v / t1), $MachinePrecision] * N[(u * -2.0), $MachinePrecision] + v), $MachinePrecision] / (-t1)), $MachinePrecision], If[LessEqual[t1, 2.1e+121], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.2000000000000002e122

   1. Initial program 44.1%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. distribute-neg-in N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 94.6

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

   5. Simplified 94.6%

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

   if -9.2000000000000002e122 < t1 < 2.1000000000000002e121

   1. Initial program 82.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 84.6

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

   4. Applied egg-rr 84.6%

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

   if 2.1000000000000002e121 < t1

   1. Initial program 48.3%

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

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 89.6%

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

      1. lift-neg.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 89.6

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

   9. Applied egg-rr 89.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u \cdot -2, v\right)}{-t1}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ t_2 := \frac{v}{t\_1}\\ \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)) (t_2 (/ v t_1)))
   (if (<= t1 -9.5e+138)
     t_2
     (if (<= t1 2.1e+121) (* v (/ t1 (* (+ t1 u) t_1))) t_2))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -9.5e+138) {
		tmp = t_2;
	} else if (t1 <= 2.1e+121) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = -t1 - u
    t_2 = v / t_1
    if (t1 <= (-9.5d+138)) then
        tmp = t_2
    else if (t1 <= 2.1d+121) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -9.5e+138) {
		tmp = t_2;
	} else if (t1 <= 2.1e+121) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 - u
	t_2 = v / t_1
	tmp = 0
	if t1 <= -9.5e+138:
		tmp = t_2
	elif t1 <= 2.1e+121:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	t_2 = Float64(v / t_1)
	tmp = 0.0
	if (t1 <= -9.5e+138)
		tmp = t_2;
	elseif (t1 <= 2.1e+121)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 - u;
	t_2 = v / t_1;
	tmp = 0.0;
	if (t1 <= -9.5e+138)
		tmp = t_2;
	elseif (t1 <= 2.1e+121)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, Block[{t$95$2 = N[(v / t$95$1), $MachinePrecision]}, If[LessEqual[t1, -9.5e+138], t$95$2, If[LessEqual[t1, 2.1e+121], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.49999999999999998e138 or 2.1000000000000002e121 < t1

   1. Initial program 45.6%

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

      1. lift-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 egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 92.3%

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

      1. lift-neg.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 92.3

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

   9. Applied egg-rr 92.3%

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

   if -9.49999999999999998e138 < t1 < 2.1000000000000002e121

   1. Initial program 82.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 84.4

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

   4. Applied egg-rr 84.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 0.215:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -1.6e-33) t_1 (if (<= t1 0.215) (* v (/ t1 (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -1.6e-33) {
		tmp = t_1;
	} else if (t1 <= 0.215) {
		tmp = v * (t1 / (u * -u));
	} 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 / (-t1 - u)
    if (t1 <= (-1.6d-33)) then
        tmp = t_1
    else if (t1 <= 0.215d0) then
        tmp = v * (t1 / (u * -u))
    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 / (-t1 - u);
	double tmp;
	if (t1 <= -1.6e-33) {
		tmp = t_1;
	} else if (t1 <= 0.215) {
		tmp = v * (t1 / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	tmp = 0
	if t1 <= -1.6e-33:
		tmp = t_1
	elif t1 <= 0.215:
		tmp = v * (t1 / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -1.6e-33)
		tmp = t_1;
	elseif (t1 <= 0.215)
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -1.6e-33)
		tmp = t_1;
	elseif (t1 <= 0.215)
		tmp = v * (t1 / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.6e-33], t$95$1, If[LessEqual[t1, 0.215], N[(v * N[(t1 / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999988e-33 or 0.214999999999999997 < t1

   1. Initial program 63.7%

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

      1. lift-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 egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 85.7%

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

      1. lift-neg.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 85.7

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

   9. Applied egg-rr 85.7%

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

   if -1.59999999999999988e-33 < t1 < 0.214999999999999997

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 71.3

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

   5. Simplified 71.3%

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

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

      6. *-commutative N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 72.2

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 0.215:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 0.215:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -1.6e-33) t_1 (if (<= t1 0.215) (* t1 (/ v (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -1.6e-33) {
		tmp = t_1;
	} else if (t1 <= 0.215) {
		tmp = t1 * (v / (u * -u));
	} 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 / (-t1 - u)
    if (t1 <= (-1.6d-33)) then
        tmp = t_1
    else if (t1 <= 0.215d0) then
        tmp = t1 * (v / (u * -u))
    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 / (-t1 - u);
	double tmp;
	if (t1 <= -1.6e-33) {
		tmp = t_1;
	} else if (t1 <= 0.215) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	tmp = 0
	if t1 <= -1.6e-33:
		tmp = t_1
	elif t1 <= 0.215:
		tmp = t1 * (v / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -1.6e-33)
		tmp = t_1;
	elseif (t1 <= 0.215)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -1.6e-33)
		tmp = t_1;
	elseif (t1 <= 0.215)
		tmp = t1 * (v / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.6e-33], t$95$1, If[LessEqual[t1, 0.215], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999988e-33 or 0.214999999999999997 < t1

   1. Initial program 63.7%

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

      1. lift-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 egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 85.7%

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

      1. lift-neg.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 85.7

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

   9. Applied egg-rr 85.7%

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

   if -1.59999999999999988e-33 < t1 < 0.214999999999999997

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 72.1

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

   5. Simplified 72.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 0.215:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.8× speedup

Math

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

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(t1 * Float64(v / Float64(t1 + u))) / Float64(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 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. lift-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 97.9

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

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

Alternative 7: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-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 97.8

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

4. Applied egg-rr 97.8%

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

5. Final simplification 97.8%

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

Alternative 8: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -5.2 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v (* u u)))))
   (if (<= u -5.2e+45) t_1 (if (<= u 1.55e+45) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * u));
	double tmp;
	if (u <= -5.2e+45) {
		tmp = t_1;
	} else if (u <= 1.55e+45) {
		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 = t1 * (v / (u * u))
    if (u <= (-5.2d+45)) then
        tmp = t_1
    else if (u <= 1.55d+45) 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 = t1 * (v / (u * u));
	double tmp;
	if (u <= -5.2e+45) {
		tmp = t_1;
	} else if (u <= 1.55e+45) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (v / (u * u))
	tmp = 0
	if u <= -5.2e+45:
		tmp = t_1
	elif u <= 1.55e+45:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -5.2e+45)
		tmp = t_1;
	elseif (u <= 1.55e+45)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (v / (u * u));
	tmp = 0.0;
	if (u <= -5.2e+45)
		tmp = t_1;
	elseif (u <= 1.55e+45)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.2e+45], t$95$1, If[LessEqual[u, 1.55e+45], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.20000000000000014e45 or 1.54999999999999994e45 < u

   1. Initial program 75.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 71.0

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

   5. Simplified 71.0%

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.6

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

   7. Applied egg-rr 73.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 79.5

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

   9. Applied egg-rr 79.5%

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

   11. Applied egg-rr 65.4%

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

   if -5.20000000000000014e45 < u < 1.54999999999999994e45

   1. Initial program 69.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 77.4

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

   5. Simplified 77.4%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+45}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+154) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+154) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.8d+154)) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+154) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e+154:
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+154)
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e+154)
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+154], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.7999999999999998e154

   1. Initial program 86.4%

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

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

   4. Applied egg-rr 100.0%

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

   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 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   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 37.4

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

   10. Simplified 37.4%

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

   if -3.7999999999999998e154 < u

   1. Initial program 69.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.6

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

   5. Simplified 61.6%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-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 98.5

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 61.4%

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

   1. lift-neg.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-+.f64 N/A

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

   4. frac-2neg N/A

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

   5. lower-/.f64 61.4

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

9. Applied egg-rr 61.4%

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

10. Final simplification 61.4%

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

Alternative 11: 16.8% 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 71.6%

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

   1. lift-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 97.9

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

4. Applied egg-rr 97.9%

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

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 61.0

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

7. Simplified 61.0%

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

8. Taylor expanded in t1 around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
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 16.4

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

10. Simplified 16.4%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))