Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.2%

Time: 10.1s

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: 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: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. lift-/.f64 N/A

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

   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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   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

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

   14. lower-/.f64 98.6

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.6

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{t1}{u - t1} \cdot \frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 6.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{-t1}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.3e+139)
   (* (/ t1 (- u t1)) (/ v t1))
   (if (<= t1 6.9e+168) (* (/ (- t1) (* (+ u t1) (+ u t1))) v) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.3e+139) {
		tmp = (t1 / (u - t1)) * (v / t1);
	} else if (t1 <= 6.9e+168) {
		tmp = (-t1 / ((u + t1) * (u + t1))) * v;
	} 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 (t1 <= (-3.3d+139)) then
        tmp = (t1 / (u - t1)) * (v / t1)
    else if (t1 <= 6.9d+168) then
        tmp = (-t1 / ((u + t1) * (u + t1))) * v
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.3e+139) {
		tmp = (t1 / (u - t1)) * (v / t1);
	} else if (t1 <= 6.9e+168) {
		tmp = (-t1 / ((u + t1) * (u + t1))) * v;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.3e+139:
		tmp = (t1 / (u - t1)) * (v / t1)
	elif t1 <= 6.9e+168:
		tmp = (-t1 / ((u + t1) * (u + t1))) * v
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.3e+139)
		tmp = Float64(Float64(t1 / Float64(u - t1)) * Float64(v / t1));
	elseif (t1 <= 6.9e+168)
		tmp = Float64(Float64(Float64(-t1) / Float64(Float64(u + t1) * Float64(u + t1))) * v);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -3.3e+139)
		tmp = (t1 / (u - t1)) * (v / t1);
	elseif (t1 <= 6.9e+168)
		tmp = (-t1 / ((u + t1) * (u + t1))) * v;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.3e+139], N[(N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.9e+168], N[(N[((-t1) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.3000000000000002e139

   1. Initial program 32.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 43.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 43.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 43.5

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

   4. Applied rewrites 43.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. lift-neg.f64 N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 94.6

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

   6. Applied rewrites 94.6%

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in u around 0

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

       1. lower-/.f64 90.3

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

   11. Applied rewrites 90.3%

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

   if -3.3000000000000002e139 < t1 < 6.8999999999999998e168

   1. Initial program 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 88.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 88.0

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

   4. Applied rewrites 88.0%

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

   if 6.8999999999999998e168 < t1

   1. Initial program 36.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{t1}{u - t1} \cdot \frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 6.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{-t1}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{-t1}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)))
   (if (<= t1 -1.35e+139)
     t_1
     (if (<= t1 6.9e+168) (* (/ (- t1) (* (+ u t1) (+ u t1))) v) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double tmp;
	if (t1 <= -1.35e+139) {
		tmp = t_1;
	} else if (t1 <= 6.9e+168) {
		tmp = (-t1 / ((u + t1) * (u + t1))) * v;
	} 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
    if (t1 <= (-1.35d+139)) then
        tmp = t_1
    else if (t1 <= 6.9d+168) then
        tmp = (-t1 / ((u + t1) * (u + t1))) * v
    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;
	double tmp;
	if (t1 <= -1.35e+139) {
		tmp = t_1;
	} else if (t1 <= 6.9e+168) {
		tmp = (-t1 / ((u + t1) * (u + t1))) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	tmp = 0
	if t1 <= -1.35e+139:
		tmp = t_1
	elif t1 <= 6.9e+168:
		tmp = (-t1 / ((u + t1) * (u + t1))) * v
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	tmp = 0.0
	if (t1 <= -1.35e+139)
		tmp = t_1;
	elseif (t1 <= 6.9e+168)
		tmp = Float64(Float64(Float64(-t1) / Float64(Float64(u + t1) * Float64(u + t1))) * v);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	tmp = 0.0;
	if (t1 <= -1.35e+139)
		tmp = t_1;
	elseif (t1 <= 6.9e+168)
		tmp = (-t1 / ((u + t1) * (u + t1))) * v;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, If[LessEqual[t1, -1.35e+139], t$95$1, If[LessEqual[t1, 6.9e+168], N[(N[((-t1) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.3499999999999999e139 or 6.8999999999999998e168 < t1

   1. Initial program 34.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 89.7

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

   5. Applied rewrites 89.7%

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

   if -1.3499999999999999e139 < t1 < 6.8999999999999998e168

   1. Initial program 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 88.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 88.0

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

   4. Applied rewrites 88.0%

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

Alternative 4: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ \mathbf{if}\;t1 \leq -4.9 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)))
   (if (<= t1 -4.9e+113)
     t_1
     (if (<= t1 6.9e+168) (* (/ (- v) (* (+ u t1) (+ u t1))) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double tmp;
	if (t1 <= -4.9e+113) {
		tmp = t_1;
	} else if (t1 <= 6.9e+168) {
		tmp = (-v / ((u + t1) * (u + t1))) * 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 / t1
    if (t1 <= (-4.9d+113)) then
        tmp = t_1
    else if (t1 <= 6.9d+168) then
        tmp = (-v / ((u + t1) * (u + t1))) * 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 / t1;
	double tmp;
	if (t1 <= -4.9e+113) {
		tmp = t_1;
	} else if (t1 <= 6.9e+168) {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	tmp = 0
	if t1 <= -4.9e+113:
		tmp = t_1
	elif t1 <= 6.9e+168:
		tmp = (-v / ((u + t1) * (u + t1))) * t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	tmp = 0.0
	if (t1 <= -4.9e+113)
		tmp = t_1;
	elseif (t1 <= 6.9e+168)
		tmp = Float64(Float64(Float64(-v) / Float64(Float64(u + t1) * Float64(u + t1))) * t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	tmp = 0.0;
	if (t1 <= -4.9e+113)
		tmp = t_1;
	elseif (t1 <= 6.9e+168)
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, If[LessEqual[t1, -4.9e+113], t$95$1, If[LessEqual[t1, 6.9e+168], N[(N[((-v) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.90000000000000021e113 or 6.8999999999999998e168 < t1

   1. Initial program 37.6%

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

   3. Taylor expanded in u around 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 87.5

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

   5. Applied rewrites 87.5%

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

   if -4.90000000000000021e113 < t1 < 6.8999999999999998e168

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 85.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.4

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

   4. Applied rewrites 85.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.9 \cdot 10^{+113}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 6.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{t1}{u \cdot u} \cdot \left(-v\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2.8e-87)
     t_1
     (if (<= t1 3.8e+21) (* (/ t1 (* u u)) (- v)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2.8e-87) {
		tmp = t_1;
	} else if (t1 <= 3.8e+21) {
		tmp = (t1 / (u * u)) * -v;
	} 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)
    if (t1 <= (-2.8d-87)) then
        tmp = t_1
    else if (t1 <= 3.8d+21) then
        tmp = (t1 / (u * u)) * -v
    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);
	double tmp;
	if (t1 <= -2.8e-87) {
		tmp = t_1;
	} else if (t1 <= 3.8e+21) {
		tmp = (t1 / (u * u)) * -v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -2.8e-87:
		tmp = t_1
	elif t1 <= 3.8e+21:
		tmp = (t1 / (u * u)) * -v
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -2.8e-87)
		tmp = t_1;
	elseif (t1 <= 3.8e+21)
		tmp = Float64(Float64(t1 / Float64(u * u)) * Float64(-v));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -2.8e-87)
		tmp = t_1;
	elseif (t1 <= 3.8e+21)
		tmp = (t1 / (u * u)) * -v;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.8e-87], t$95$1, If[LessEqual[t1, 3.8e+21], N[(N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision] * (-v)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.8000000000000001e-87 or 3.8e21 < t1

   1. Initial program 54.9%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      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

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.0

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

   7. Applied rewrites 83.0%

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

   if -2.8000000000000001e-87 < t1 < 3.8e21

   1. Initial program 84.3%

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

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{t1}{u \cdot u} \cdot \left(-v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-2neg N/A

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

   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. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 98.6

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

Alternative 7: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+220}:\\ \;\;\;\;\frac{t1}{u \cdot u} \cdot v\\ \mathbf{elif}\;u \leq 2.45 \cdot 10^{+42}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot u} \cdot t1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e+220)
   (* (/ t1 (* u u)) v)
   (if (<= u 2.45e+42) (/ (- v) (+ u t1)) (* (/ v (* u u)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+220) {
		tmp = (t1 / (u * u)) * v;
	} else if (u <= 2.45e+42) {
		tmp = -v / (u + t1);
	} else {
		tmp = (v / (u * u)) * t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.1d+220)) then
        tmp = (t1 / (u * u)) * v
    else if (u <= 2.45d+42) then
        tmp = -v / (u + t1)
    else
        tmp = (v / (u * u)) * t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+220) {
		tmp = (t1 / (u * u)) * v;
	} else if (u <= 2.45e+42) {
		tmp = -v / (u + t1);
	} else {
		tmp = (v / (u * u)) * t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e+220:
		tmp = (t1 / (u * u)) * v
	elif u <= 2.45e+42:
		tmp = -v / (u + t1)
	else:
		tmp = (v / (u * u)) * t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e+220)
		tmp = Float64(Float64(t1 / Float64(u * u)) * v);
	elseif (u <= 2.45e+42)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(v / Float64(u * u)) * t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e+220)
		tmp = (t1 / (u * u)) * v;
	elseif (u <= 2.45e+42)
		tmp = -v / (u + t1);
	else
		tmp = (v / (u * u)) * t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e+220], N[(N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], If[LessEqual[u, 2.45e+42], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.10000000000000007e220

   1. Initial program 91.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 92.2

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.2

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 92.2%

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

   if -2.10000000000000007e220 < u < 2.4500000000000001e42

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      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

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

      14. lower-/.f64 98.1

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.2

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

   7. Applied rewrites 66.2%

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

   if 2.4500000000000001e42 < u

   1. Initial program 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 64.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 64.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. +-lft-identity N/A

         \[\leadsto \frac{v}{u \cdot u} \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      9. flip3-+ N/A

         \[\leadsto \frac{v}{u \cdot u} \cdot \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}} \]

      10. sqr-pow N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + \color{blue}{{\left(\mathsf{neg}\left(t1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(t1\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      11. pow-prod-down N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + \color{blue}{{\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + {\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + {\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      14. sqr-neg N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + {\color{blue}{\left(t1 \cdot t1\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      15. pow-prod-down N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + \color{blue}{{t1}^{\left(\frac{3}{2}\right)} \cdot {t1}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

      16. sqr-pow N/A

          \[\leadsto \frac{v}{u \cdot u} \cdot \frac{{0}^{3} + \color{blue}{{t1}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

   9. Applied rewrites 54.9%

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

Alternative 8: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{u \cdot u} \cdot v\\ \mathbf{if}\;u \leq -2.1 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 (* u u)) v)))
   (if (<= u -2.1e+220) t_1 (if (<= u 2.3e+148) (/ (- v) (+ u t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / (u * u)) * v;
	double tmp;
	if (u <= -2.1e+220) {
		tmp = t_1;
	} else if (u <= 2.3e+148) {
		tmp = -v / (u + 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 / (u * u)) * v
    if (u <= (-2.1d+220)) then
        tmp = t_1
    else if (u <= 2.3d+148) then
        tmp = -v / (u + 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 / (u * u)) * v;
	double tmp;
	if (u <= -2.1e+220) {
		tmp = t_1;
	} else if (u <= 2.3e+148) {
		tmp = -v / (u + t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / (u * u)) * v
	tmp = 0
	if u <= -2.1e+220:
		tmp = t_1
	elif u <= 2.3e+148:
		tmp = -v / (u + t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / Float64(u * u)) * v)
	tmp = 0.0
	if (u <= -2.1e+220)
		tmp = t_1;
	elseif (u <= 2.3e+148)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / (u * u)) * v;
	tmp = 0.0;
	if (u <= -2.1e+220)
		tmp = t_1;
	elseif (u <= 2.3e+148)
		tmp = -v / (u + t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision]}, If[LessEqual[u, -2.1e+220], t$95$1, If[LessEqual[u, 2.3e+148], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.10000000000000007e220 or 2.3000000000000001e148 < u

   1. Initial program 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 70.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.0

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

   7. Applied rewrites 70.0%

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

   9. Applied rewrites 67.0%

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

   if -2.10000000000000007e220 < u < 2.3000000000000001e148

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      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

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

      14. lower-/.f64 98.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.9

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

   7. Applied rewrites 63.9%

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

Alternative 9: 62.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. lift-/.f64 N/A

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

   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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   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

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

   14. lower-/.f64 98.6

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.6

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 59.4

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

7. Applied rewrites 59.4%

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

Alternative 10: 54.8% 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 68.4%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 54.8

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

5. Applied rewrites 54.8%

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

Alternative 11: 14.7% accurate, 2.5× 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(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 68.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 74.6

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 74.6

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 74.6

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

4. Applied rewrites 74.6%

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

5. Taylor expanded in u around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 45.0

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

7. Applied rewrites 45.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

9. Applied rewrites 21.9%

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

10. Taylor expanded in u around 0

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

    1. lower-/.f64 13.4

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

12. Applied rewrites 13.4%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))