Rosa's DopplerBench

Percentage Accurate: 72.5% → 98.2%

Time: 7.2s

Alternatives: 14

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: 72.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 99.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+170}:\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u, -v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.02e+170)
   (* v (/ -1.0 (fma 2.0 u t1)))
   (if (<= t1 6e+127)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (fma (/ v t1) u (- v)) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.02e+170) {
		tmp = v * (-1.0 / fma(2.0, u, t1));
	} else if (t1 <= 6e+127) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = fma((v / t1), u, -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.02e+170)
		tmp = Float64(v * Float64(-1.0 / fma(2.0, u, t1)));
	elseif (t1 <= 6e+127)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(fma(Float64(v / t1), u, Float64(-v)) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.02e+170], N[(v * N[(-1.0 / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6e+127], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / t1), $MachinePrecision] * u + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.02000000000000002e170

   1. Initial program 43.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 94.2

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

   9. Applied rewrites 94.2%

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

   if -1.02000000000000002e170 < t1 < 6.0000000000000005e127

   1. Initial program 83.6%

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

   if 6.0000000000000005e127 < t1

   1. Initial program 31.2%

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

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

      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 100.0

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

      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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+170}:\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u, -v\right)}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+170} \lor \neg \left(t1 \leq 9.2 \cdot 10^{+125}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.02e+170) (not (<= t1 9.2e+125)))
   (* v (/ -1.0 (fma 2.0 u t1)))
   (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.02e+170) || !(t1 <= 9.2e+125)) {
		tmp = v * (-1.0 / fma(2.0, u, t1));
	} else {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.02e+170) || !(t1 <= 9.2e+125))
		tmp = Float64(v * Float64(-1.0 / fma(2.0, u, t1)));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.02e+170], N[Not[LessEqual[t1, 9.2e+125]], $MachinePrecision]], N[(v * N[(-1.0 / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.02000000000000002e170 or 9.20000000000000051e125 < t1

   1. Initial program 37.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.8

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

   9. Applied rewrites 96.8%

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

   if -1.02000000000000002e170 < t1 < 9.20000000000000051e125

   1. Initial program 83.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+170} \lor \neg \left(t1 \leq 9.2 \cdot 10^{+125}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+55} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+90}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e+55) (not (<= t1 1.45e+90)))
   (* v (/ -1.0 (fma 2.0 u t1)))
   (/ (* (/ v u) t1) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+55) || !(t1 <= 1.45e+90)) {
		tmp = v * (-1.0 / fma(2.0, u, t1));
	} else {
		tmp = ((v / u) * t1) / -u;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e+55) || !(t1 <= 1.45e+90))
		tmp = Float64(v * Float64(-1.0 / fma(2.0, u, t1)));
	else
		tmp = Float64(Float64(Float64(v / u) * t1) / Float64(-u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e+55], N[Not[LessEqual[t1, 1.45e+90]], $MachinePrecision]], N[(v * N[(-1.0 / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / u), $MachinePrecision] * t1), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.49999999999999998e55 or 1.4500000000000001e90 < 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

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if -4.49999999999999998e55 < t1 < 1.4500000000000001e90

   1. Initial program 81.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 74.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+55} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+90}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+55} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+90}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e+55) (not (<= t1 1.45e+90)))
   (* v (/ -1.0 (fma 2.0 u t1)))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+55) || !(t1 <= 1.45e+90)) {
		tmp = v * (-1.0 / fma(2.0, u, t1));
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e+55) || !(t1 <= 1.45e+90))
		tmp = Float64(v * Float64(-1.0 / fma(2.0, u, t1)));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e+55], N[Not[LessEqual[t1, 1.45e+90]], $MachinePrecision]], N[(v * N[(-1.0 / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.49999999999999998e55 or 1.4500000000000001e90 < 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

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if -4.49999999999999998e55 < t1 < 1.4500000000000001e90

   1. Initial program 81.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+55} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+90}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.42e-10) (not (<= t1 8.5e+31)))
   (* v (/ -1.0 (fma 2.0 u t1)))
   (* v (/ (/ (- t1) u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = v * (-1.0 / fma(2.0, u, t1));
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31))
		tmp = Float64(v * Float64(-1.0 / fma(2.0, u, t1)));
	else
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.42e-10], N[Not[LessEqual[t1, 8.5e+31]], $MachinePrecision]], N[(v * N[(-1.0 / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.42000000000000001e-10 or 8.49999999999999947e31 < t1

   1. Initial program 59.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.4

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

   6. Applied rewrites 97.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 85.3

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

   9. Applied rewrites 85.3%

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

   if -1.42000000000000001e-10 < t1 < 8.49999999999999947e31

   1. Initial program 83.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 71.1%

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

   9. Applied rewrites 75.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.42e-10) (not (<= t1 8.5e+31)))
   (* v (/ -1.0 (fma 2.0 u t1)))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = v * (-1.0 / fma(2.0, u, t1));
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31))
		tmp = Float64(v * Float64(-1.0 / fma(2.0, u, t1)));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.42e-10], N[Not[LessEqual[t1, 8.5e+31]], $MachinePrecision]], N[(v * N[(-1.0 / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.42000000000000001e-10 or 8.49999999999999947e31 < t1

   1. Initial program 59.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.4

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

   6. Applied rewrites 97.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 85.3

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

   9. Applied rewrites 85.3%

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

   if -1.42000000000000001e-10 < t1 < 8.49999999999999947e31

   1. Initial program 83.0%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;v \cdot \frac{-1}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.42e-10) (not (<= t1 8.5e+31)))
   (/ (- v) (+ u t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * 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 ((t1 <= (-1.42d-10)) .or. (.not. (t1 <= 8.5d+31))) then
        tmp = -v / (u + t1)
    else
        tmp = (-t1 * v) / (u * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.42e-10) or not (t1 <= 8.5e+31):
		tmp = -v / (u + t1)
	else:
		tmp = (-t1 * v) / (u * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.42e-10) || ~((t1 <= 8.5e+31)))
		tmp = -v / (u + t1);
	else
		tmp = (-t1 * v) / (u * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.42e-10], N[Not[LessEqual[t1, 8.5e+31]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.42000000000000001e-10 or 8.49999999999999947e31 < t1

   1. Initial program 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   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 84.6

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

   7. Applied rewrites 84.6%

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

   if -1.42000000000000001e-10 < t1 < 8.49999999999999947e31

   1. Initial program 83.0%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(-u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.42e-10) (not (<= t1 8.5e+31)))
   (/ (- v) (+ u t1))
   (* v (/ t1 (* (- u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = -v / (u + t1);
	} else {
		tmp = v * (t1 / (-u * 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 ((t1 <= (-1.42d-10)) .or. (.not. (t1 <= 8.5d+31))) then
        tmp = -v / (u + t1)
    else
        tmp = v * (t1 / (-u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = -v / (u + t1);
	} else {
		tmp = v * (t1 / (-u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.42e-10) or not (t1 <= 8.5e+31):
		tmp = -v / (u + t1)
	else:
		tmp = v * (t1 / (-u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(v * Float64(t1 / Float64(Float64(-u) * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.42e-10) || ~((t1 <= 8.5e+31)))
		tmp = -v / (u + t1);
	else
		tmp = v * (t1 / (-u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.42e-10], N[Not[LessEqual[t1, 8.5e+31]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.42000000000000001e-10 or 8.49999999999999947e31 < t1

   1. Initial program 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   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 84.6

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

   7. Applied rewrites 84.6%

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

   if -1.42000000000000001e-10 < t1 < 8.49999999999999947e31

   1. Initial program 83.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 71.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(-u\right) \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.42e-10) (not (<= t1 8.5e+31)))
   (/ (- v) (+ u t1))
   (* t1 (/ v (* (- u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = -v / (u + t1);
	} else {
		tmp = t1 * (v / (-u * 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 ((t1 <= (-1.42d-10)) .or. (.not. (t1 <= 8.5d+31))) then
        tmp = -v / (u + t1)
    else
        tmp = t1 * (v / (-u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31)) {
		tmp = -v / (u + t1);
	} else {
		tmp = t1 * (v / (-u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.42e-10) or not (t1 <= 8.5e+31):
		tmp = -v / (u + t1)
	else:
		tmp = t1 * (v / (-u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.42e-10) || !(t1 <= 8.5e+31))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(t1 * Float64(v / Float64(Float64(-u) * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.42e-10) || ~((t1 <= 8.5e+31)))
		tmp = -v / (u + t1);
	else
		tmp = t1 * (v / (-u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.42e-10], N[Not[LessEqual[t1, 8.5e+31]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.42000000000000001e-10 or 8.49999999999999947e31 < t1

   1. Initial program 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   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 84.6

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

   7. Applied rewrites 84.6%

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

   if -1.42000000000000001e-10 < t1 < 8.49999999999999947e31

   1. Initial program 83.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 69.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.42 \cdot 10^{-10} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 71.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   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

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 97.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 97.5

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

4. Applied rewrites 97.5%

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

Alternative 12: 95.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   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

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 97.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 97.5

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

4. Applied rewrites 97.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 95.6

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 95.6

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 95.6

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

6. Applied rewrites 95.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 95.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 95.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 95.8

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

8. Applied rewrites 95.8%

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

Alternative 13: 61.0% 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 71.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 99.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

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 59.2

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

7. Applied rewrites 59.2%

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

Alternative 14: 53.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 51.9

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

5. Applied rewrites 51.9%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))