Rosa's DopplerBench

Percentage Accurate: 72.2% → 98.1%

Time: 7.3s

Alternatives: 12

Speedup: 0.9×

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: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

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

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 76.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot t1\\ \mathbf{if}\;t1 \leq -7 \cdot 10^{+99}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq -9.2 \cdot 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) (* (+ t1 u) (+ t1 u))) t1)))
   (if (<= t1 -7e+99)
     (/ (- v) (fma 2.0 u t1))
     (if (<= t1 -9.2e-105)
       t_1
       (if (<= t1 8.5e-149)
         (/ (* (/ t1 u) v) (- u))
         (if (<= t1 1.6e+84) t_1 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / ((t1 + u) * (t1 + u))) * t1;
	double tmp;
	if (t1 <= -7e+99) {
		tmp = -v / fma(2.0, u, t1);
	} else if (t1 <= -9.2e-105) {
		tmp = t_1;
	} else if (t1 <= 8.5e-149) {
		tmp = ((t1 / u) * v) / -u;
	} else if (t1 <= 1.6e+84) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(t1 + u))) * t1)
	tmp = 0.0
	if (t1 <= -7e+99)
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	elseif (t1 <= -9.2e-105)
		tmp = t_1;
	elseif (t1 <= 8.5e-149)
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	elseif (t1 <= 1.6e+84)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]}, If[LessEqual[t1, -7e+99], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -9.2e-105], t$95$1, If[LessEqual[t1, 8.5e-149], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 1.6e+84], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -6.9999999999999995e99

   1. Initial program 55.2%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 96.3%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 90.0

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

   12. Applied rewrites 90.0%

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

   if -6.9999999999999995e99 < t1 < -9.2000000000000004e-105 or 8.5000000000000006e-149 < t1 < 1.60000000000000005e84

   1. Initial program 89.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.8

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t1 around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 95.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*r/ N/A

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

       5. associate-/r/ N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 94.2

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

   11. Applied rewrites 94.2%

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

   if -9.2000000000000004e-105 < t1 < 8.5000000000000006e-149

   1. Initial program 84.5%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 86.5%

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

   if 1.60000000000000005e84 < t1

   1. Initial program 53.3%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+99}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq -9.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot t1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.5e-92) (not (<= t1 9.2e-21)))
   (/ (- v) (fma 2.0 u t1))
   (/ (* (/ t1 u) v) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.5e-92) || !(t1 <= 9.2e-21)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.5e-92) || !(t1 <= 9.2e-21))
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	else
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.5e-92], N[Not[LessEqual[t1, 9.2e-21]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.5000000000000005e-92 or 9.19999999999999998e-21 < t1

   1. Initial program 69.2%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 96.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 85.7

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

   12. Applied rewrites 85.7%

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

   if -7.5000000000000005e-92 < t1 < 9.19999999999999998e-21

   1. Initial program 85.6%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 81.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.5e-92) (not (<= t1 9.2e-21)))
   (/ (- v) (fma 2.0 u t1))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.5e-92) || !(t1 <= 9.2e-21)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.5e-92) || !(t1 <= 9.2e-21))
		tmp = Float64(Float64(-v) / 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, -7.5e-92], N[Not[LessEqual[t1, 9.2e-21]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.5000000000000005e-92 or 9.19999999999999998e-21 < t1

   1. Initial program 69.2%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 96.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 85.7

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

   12. Applied rewrites 85.7%

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

   if -7.5000000000000005e-92 < t1 < 9.19999999999999998e-21

   1. Initial program 85.6%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+85}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.02e+85)
   (/ (- v) (fma 2.0 u t1))
   (if (<= t1 2.5e+86) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.02e+85) {
		tmp = -v / fma(2.0, u, t1);
	} else if (t1 <= 2.5e+86) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.02e+85)
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	elseif (t1 <= 2.5e+86)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.02e+85], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.5e+86], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.02e85

   1. Initial program 57.3%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 96.6%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 91.0

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

   12. Applied rewrites 91.0%

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

   if -1.02e85 < t1 < 2.4999999999999999e86

   1. Initial program 87.5%

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

   if 2.4999999999999999e86 < t1

   1. Initial program 51.3%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 6: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.5e-92) (not (<= t1 9.2e-21)))
   (/ (- v) (fma 2.0 u t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.5e-92) || !(t1 <= 9.2e-21)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.5e-92) || !(t1 <= 9.2e-21))
		tmp = Float64(Float64(-v) / 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, -7.5e-92], N[Not[LessEqual[t1, 9.2e-21]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.5000000000000005e-92 or 9.19999999999999998e-21 < t1

   1. Initial program 69.2%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 96.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 85.7

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

   12. Applied rewrites 85.7%

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

   if -7.5000000000000005e-92 < t1 < 9.19999999999999998e-21

   1. Initial program 85.6%

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

   3. Taylor expanded in u around 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 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7e-92) (not (<= t1 9.2e-21)))
   (/ (- v) (fma 2.0 u t1))
   (* v (/ (- t1) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e-92) || !(t1 <= 9.2e-21)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7e-92) || !(t1 <= 9.2e-21))
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7e-92], N[Not[LessEqual[t1, 9.2e-21]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7e-92 or 9.19999999999999998e-21 < t1

   1. Initial program 69.2%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 96.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 85.7

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

   12. Applied rewrites 85.7%

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

   if -7e-92 < t1 < 9.19999999999999998e-21

   1. Initial program 85.6%

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

   3. Taylor expanded in u around 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 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 72.6

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

   7. Applied rewrites 72.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{-92} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% 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 76.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.7

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

   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 97.7

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

4. Applied rewrites 97.7%

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

Alternative 9: 94.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in t1 around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. lower-*.f64 N/A

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

9. Applied rewrites 95.0%

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

10. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 N/A

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

    6. lower-/.f64 95.0

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

12. Applied rewrites 95.0%

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

Alternative 10: 62.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in t1 around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. lower-*.f64 N/A

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

9. Applied rewrites 95.0%

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

10. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 65.4

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

12. Applied rewrites 65.4%

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

Alternative 11: 61.7% 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 76.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

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 65.0

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

7. Applied rewrites 65.0%

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

Alternative 12: 54.5% 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 76.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 57.2

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

5. Applied rewrites 57.2%

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

Reproduce

herbie shell --seed 2024302 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))