Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.1%

Time: 7.4s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 97.8

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 97.8

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

4. Applied rewrites 97.8%

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

5. Final simplification 97.8%

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

Alternative 2: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.45 \cdot 10^{+118}:\\ \;\;\;\;\frac{\left(\frac{-1}{u} \cdot v\right) \cdot t1}{u}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{+202}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2 - \frac{u}{t1}, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.45e+118)
   (/ (* (* (/ -1.0 u) v) t1) u)
   (if (<= u 3.6e+202)
     (/ v (fma (- -2.0 (/ u t1)) u (- t1)))
     (/ (* (/ (- v) u) t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.45e+118) {
		tmp = (((-1.0 / u) * v) * t1) / u;
	} else if (u <= 3.6e+202) {
		tmp = v / fma((-2.0 - (u / t1)), u, -t1);
	} else {
		tmp = ((-v / u) * t1) / u;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.45e+118)
		tmp = Float64(Float64(Float64(Float64(-1.0 / u) * v) * t1) / u);
	elseif (u <= 3.6e+202)
		tmp = Float64(v / fma(Float64(-2.0 - Float64(u / t1)), u, Float64(-t1)));
	else
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.45e+118], N[(N[(N[(N[(-1.0 / u), $MachinePrecision] * v), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 3.6e+202], N[(v / N[(N[(-2.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.45000000000000001e118

   1. Initial program 73.1%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 75.7%

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

   9. Applied rewrites 93.5%

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

   11. Applied rewrites 93.6%

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

   if -3.45000000000000001e118 < u < 3.60000000000000008e202

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 97.4

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 66.8

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

   7. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 66.8

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. mul-1-neg N/A

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

       8. unsub-neg N/A

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

       9. lower--.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 97.4

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

   12. Applied rewrites 97.4%

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

   if 3.60000000000000008e202 < u

   1. Initial program 61.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 95.3%

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

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -1.36 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -1.36e+78)
     t_1
     (if (<= t1 1.6e+81) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -1.36e+78) {
		tmp = t_1;
	} else if (t1 <= 1.6e+81) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -1.36e+78)
		tmp = t_1;
	elseif (t1 <= 1.6e+81)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.36e+78], t$95$1, If[LessEqual[t1, 1.6e+81], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.35999999999999999e78 or 1.6e81 < t1

   1. Initial program 49.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 96.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 96.2

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 87.7

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

   7. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 87.7

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

   9. Applied rewrites 87.7%

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

   if -1.35999999999999999e78 < t1 < 1.6e81

   1. Initial program 83.3%

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

Alternative 4: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{t1}{\frac{-u}{v} \cdot u}\\ \mathbf{elif}\;u \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.6e-37)
   (/ t1 (* (/ (- u) v) u))
   (if (<= u 5.5e+57) (/ (- v) t1) (/ (* (/ (- v) u) t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.6e-37) {
		tmp = t1 / ((-u / v) * u);
	} else if (u <= 5.5e+57) {
		tmp = -v / t1;
	} else {
		tmp = ((-v / u) * t1) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.6d-37)) then
        tmp = t1 / ((-u / v) * u)
    else if (u <= 5.5d+57) then
        tmp = -v / t1
    else
        tmp = ((-v / u) * t1) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.6e-37) {
		tmp = t1 / ((-u / v) * u);
	} else if (u <= 5.5e+57) {
		tmp = -v / t1;
	} else {
		tmp = ((-v / u) * t1) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.6e-37:
		tmp = t1 / ((-u / v) * u)
	elif u <= 5.5e+57:
		tmp = -v / t1
	else:
		tmp = ((-v / u) * t1) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.6e-37)
		tmp = Float64(t1 / Float64(Float64(Float64(-u) / v) * u));
	elseif (u <= 5.5e+57)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.6e-37)
		tmp = t1 / ((-u / v) * u);
	elseif (u <= 5.5e+57)
		tmp = -v / t1;
	else
		tmp = ((-v / u) * t1) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.6e-37], N[(t1 / N[(N[((-u) / v), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5.5e+57], N[((-v) / t1), $MachinePrecision], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.5999999999999999e-37

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. frac-2neg N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 91.6

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 83.7

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

   7. Applied rewrites 83.7%

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

   if -1.5999999999999999e-37 < u < 5.5000000000000002e57

   1. Initial program 65.4%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if 5.5000000000000002e57 < u

   1. Initial program 72.5%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 83.7%

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

Alternative 5: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{if}\;u \leq -1.12 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (/ (- v) u) t1) u)))
   (if (<= u -1.12e-64) t_1 (if (<= u 5.5e+57) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = ((-v / u) * t1) / u;
	double tmp;
	if (u <= -1.12e-64) {
		tmp = t_1;
	} else if (u <= 5.5e+57) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-v / u) * t1) / u
    if (u <= (-1.12d-64)) then
        tmp = t_1
    else if (u <= 5.5d+57) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = ((-v / u) * t1) / u;
	double tmp;
	if (u <= -1.12e-64) {
		tmp = t_1;
	} else if (u <= 5.5e+57) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = ((-v / u) * t1) / u
	tmp = 0
	if u <= -1.12e-64:
		tmp = t_1
	elif u <= 5.5e+57:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(Float64(-v) / u) * t1) / u)
	tmp = 0.0
	if (u <= -1.12e-64)
		tmp = t_1;
	elseif (u <= 5.5e+57)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = ((-v / u) * t1) / u;
	tmp = 0.0;
	if (u <= -1.12e-64)
		tmp = t_1;
	elseif (u <= 5.5e+57)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision]}, If[LessEqual[u, -1.12e-64], t$95$1, If[LessEqual[u, 5.5e+57], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.12e-64 or 5.5000000000000002e57 < u

   1. Initial program 76.0%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 82.6%

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

   if -1.12e-64 < u < 5.5000000000000002e57

   1. Initial program 64.3%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.9

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

   5. Applied rewrites 79.9%

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

Alternative 6: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -1.4e-62)
     t_1
     (if (<= t1 1.9e+26) (/ (* (/ (- t1) u) v) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -1.4e-62) {
		tmp = t_1;
	} else if (t1 <= 1.9e+26) {
		tmp = ((-t1 / u) * v) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -1.4e-62)
		tmp = t_1;
	elseif (t1 <= 1.9e+26)
		tmp = Float64(Float64(Float64(Float64(-t1) / u) * v) / u);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.4e-62], t$95$1, If[LessEqual[t1, 1.9e+26], N[(N[(N[((-t1) / u), $MachinePrecision] * v), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.40000000000000001e-62 or 1.9000000000000001e26 < t1

   1. Initial program 58.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 93.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 82.6

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

   9. Applied rewrites 82.6%

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

   if -1.40000000000000001e-62 < t1 < 1.9000000000000001e26

   1. Initial program 83.4%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 77.7%

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

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -1.4e-62)
     t_1
     (if (<= t1 4.4e-37) (/ (* (- t1) v) (* u u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -1.4e-62) {
		tmp = t_1;
	} else if (t1 <= 4.4e-37) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -1.4e-62)
		tmp = t_1;
	elseif (t1 <= 4.4e-37)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.4e-62], t$95$1, If[LessEqual[t1, 4.4e-37], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.40000000000000001e-62 or 4.40000000000000004e-37 < t1

   1. Initial program 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 93.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 80.2

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

   7. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 80.2

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

   9. Applied rewrites 80.2%

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

   if -1.40000000000000001e-62 < t1 < 4.40000000000000004e-37

   1. Initial program 83.3%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

Alternative 8: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -1.4e-62)
     t_1
     (if (<= t1 4.4e-37) (* (/ v (* (- u) u)) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -1.4e-62) {
		tmp = t_1;
	} else if (t1 <= 4.4e-37) {
		tmp = (v / (-u * u)) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -1.4e-62)
		tmp = t_1;
	elseif (t1 <= 4.4e-37)
		tmp = Float64(Float64(v / Float64(Float64(-u) * u)) * t1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.4e-62], t$95$1, If[LessEqual[t1, 4.4e-37], N[(N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.40000000000000001e-62 or 4.40000000000000004e-37 < t1

   1. Initial program 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 93.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 80.2

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

   7. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 80.2

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

   9. Applied rewrites 80.2%

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

   if -1.40000000000000001e-62 < t1 < 4.40000000000000004e-37

   1. Initial program 83.3%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 77.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 0.8× speedup

Math

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

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 70.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 97.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 97.5

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

4. Applied rewrites 97.5%

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

5. Final simplification 97.5%

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

Alternative 10: 56.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{-2 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.35e+118) (/ v (* -2.0 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.35e+118) {
		tmp = v / (-2.0 * u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.35d+118)) then
        tmp = v / ((-2.0d0) * u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.35e+118) {
		tmp = v / (-2.0 * u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.35e+118:
		tmp = v / (-2.0 * u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.35e+118)
		tmp = Float64(v / Float64(-2.0 * u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.35e+118)
		tmp = v / (-2.0 * u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.35e+118], N[(v / N[(-2.0 * u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.35e118

   1. Initial program 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 83.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 83.2

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 44.0

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

   7. Applied rewrites 44.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 44.0

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

   9. Applied rewrites 44.0%

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

   10. Taylor expanded in u around inf

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

   12. Applied rewrites 37.7%

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

   if -1.35e118 < u

   1. Initial program 69.5%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 60.8

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

   5. Applied rewrites 60.8%

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

Alternative 11: 62.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 92.9

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 92.9

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 92.9

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

4. Applied rewrites 92.9%

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

5. Taylor expanded in u around 0

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

   1. lower-fma.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 60.4

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

7. Applied rewrites 60.4%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 60.4

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

9. Applied rewrites 60.4%

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

Alternative 12: 54.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 53.9

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

5. Applied rewrites 53.9%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))