Rosa's DopplerBench

Percentage Accurate: 72.8% → 97.7%

Time: 7.2s

Alternatives: 13

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: 72.8% 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: 97.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. frac-2neg N/A

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

   14. lift-neg.f64 N/A

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

   15. remove-double-neg N/A

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

   16. lower-/.f64 N/A

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

   17. lower-neg.f64 98.4

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{+94}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(\frac{u}{t1}, t1, t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.25e+94)
   (/ v (fma -2.0 u (- t1)))
   (if (<= t1 5.2e+77)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (- v) (fma (/ u t1) t1 t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.25e+94) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (t1 <= 5.2e+77) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / fma((u / t1), t1, t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.25e+94)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (t1 <= 5.2e+77)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / fma(Float64(u / t1), t1, t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.25e+94], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.2e+77], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(N[(u / t1), $MachinePrecision] * t1 + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.25000000000000003e94

   1. Initial program 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 92.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 92.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 92.8

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

   4. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 92.8

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

   6. Applied rewrites 92.8%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 85.5

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

   9. Applied rewrites 85.5%

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

   if -1.25000000000000003e94 < t1 < 5.2000000000000004e77

   1. Initial program 89.5%

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

   if 5.2000000000000004e77 < t1

   1. Initial program 46.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   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 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in t1 around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.8

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

   10. Applied rewrites 89.8%

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

Alternative 3: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{+94}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.25e+94)
   (/ v (fma -2.0 u (- t1)))
   (if (<= t1 5.2e+77)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.25e+94) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (t1 <= 5.2e+77) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.25e+94)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (t1 <= 5.2e+77)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.25e+94], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.2e+77], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.25000000000000003e94

   1. Initial program 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 92.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 92.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 92.8

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

   4. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 92.8

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

   6. Applied rewrites 92.8%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 85.5

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

   9. Applied rewrites 85.5%

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

   if -1.25000000000000003e94 < t1 < 5.2000000000000004e77

   1. Initial program 89.5%

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

   if 5.2000000000000004e77 < t1

   1. Initial program 46.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   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 89.8

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

   7. Applied rewrites 89.8%

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

Alternative 4: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.1e-8)
   (/ v (fma -2.0 u (- t1)))
   (if (<= t1 2.15e-81) (* (/ t1 u) (/ (- v) u)) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.1e-8) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (t1 <= 2.15e-81) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.1e-8)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (t1 <= 2.15e-81)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.1e-8], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.15e-81], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.0999999999999999e-8

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 93.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 93.8

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if -1.0999999999999999e-8 < t1 < 2.15000000000000015e-81

   1. Initial program 89.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   if 2.15000000000000015e-81 < t1

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.9

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

   7. Applied rewrites 77.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-81}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.1e-8)
   (/ v (fma -2.0 u (- t1)))
   (if (<= t1 2.15e-81) (* t1 (/ v (* (- u) u))) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.1e-8) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (t1 <= 2.15e-81) {
		tmp = t1 * (v / (-u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.1e-8)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (t1 <= 2.15e-81)
		tmp = Float64(t1 * Float64(v / Float64(Float64(-u) * u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.1e-8], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.15e-81], N[(t1 * N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.0999999999999999e-8

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 93.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 93.8

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if -1.0999999999999999e-8 < t1 < 2.15000000000000015e-81

   1. Initial program 89.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 81.9%

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

   if 2.15000000000000015e-81 < t1

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.9

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

   7. Applied rewrites 77.9%

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

Alternative 6: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-81}:\\ \;\;\;\;v \cdot \frac{t1}{\left(-u\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.1e-8)
   (/ v (fma -2.0 u (- t1)))
   (if (<= t1 2.15e-81) (* v (/ t1 (* (- u) u))) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.1e-8) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (t1 <= 2.15e-81) {
		tmp = v * (t1 / (-u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.1e-8)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (t1 <= 2.15e-81)
		tmp = Float64(v * Float64(t1 / Float64(Float64(-u) * u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.1e-8], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.15e-81], N[(v * N[(t1 / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.0999999999999999e-8

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 93.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 93.8

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if -1.0999999999999999e-8 < t1 < 2.15000000000000015e-81

   1. Initial program 89.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 81.5%

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

   if 2.15000000000000015e-81 < t1

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.9

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

   7. Applied rewrites 77.9%

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

Alternative 7: 98.1% accurate, 0.8× speedup

Math

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

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 74.8%

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

   1. lift-/.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 98.1

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

   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 98.1

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

   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 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 8: 94.9% accurate, 0.8× speedup

Math

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

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

Derivation

1. Initial program 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 96.1

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 96.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 96.1

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

4. Applied rewrites 96.1%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 96.1

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

6. Applied rewrites 96.1%

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

7. Final simplification 96.1%

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

Alternative 9: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(u + t1\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (* (- -1.0 (/ u t1)) (+ u t1))))

C

double code(double u, double v, double t1) {
	return v / ((-1.0 - (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 / (((-1.0d0) - (u / t1)) * (u + t1))
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((-1.0 - (u / t1)) * (u + t1));
}

Python

def code(u, v, t1):
	return v / ((-1.0 - (u / t1)) * (u + t1))

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(-1.0 - Float64(u / t1)) * Float64(u + t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((-1.0 - (u / t1)) * (u + t1));
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 96.1

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 96.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 96.1

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

4. Applied rewrites 96.1%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 96.1

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

6. Applied rewrites 96.1%

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

7. Taylor expanded in u around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 96.1

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

9. Applied rewrites 96.1%

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

Alternative 10: 94.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 96.1

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 96.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 96.1

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

4. Applied rewrites 96.1%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 96.1

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

6. Applied rewrites 96.1%

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

7. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 96.1

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

9. Applied rewrites 96.1%

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

Alternative 11: 62.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 96.1

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 96.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 96.1

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

4. Applied rewrites 96.1%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 96.1

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

6. Applied rewrites 96.1%

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

7. Taylor expanded in u around 0

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

   1. lower-fma.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 59.0

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

9. Applied rewrites 59.0%

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

Alternative 12: 62.1% 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 74.8%

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

   1. lift-/.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 98.1

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

   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 98.1

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

   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 98.1

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 58.5

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

7. Applied rewrites 58.5%

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

Alternative 13: 54.6% 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 74.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 50.7

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

5. Applied rewrites 50.7%

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

Reproduce

herbie shell --seed 2024314 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))