Rosa's DopplerBench

Percentage Accurate: 72.3% → 97.7%

Time: 9.6s

Alternatives: 12

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-Float64(t1 + 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 + u), $MachinePrecision]), $MachinePrecision] / N[((-N[(t1 + u), $MachinePrecision]) / t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. frac-2neg N/A

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

   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 99.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 \cdot v}{\left(-\left(t1 + u\right)\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -2.35 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{-186}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 v) (* (- (+ t1 u)) (+ t1 u)))))
   (if (<= t1 -3.5e+136)
     (/ (- v) t1)
     (if (<= t1 -2.35e-136)
       t_1
       (if (<= t1 3.6e-186)
         (* (/ t1 u) (/ (- v) u))
         (if (<= t1 2.5e+109) t_1 (/ (- v) (fma u 2.0 t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 * v) / (-(t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -3.5e+136) {
		tmp = -v / t1;
	} else if (t1 <= -2.35e-136) {
		tmp = t_1;
	} else if (t1 <= 3.6e-186) {
		tmp = (t1 / u) * (-v / u);
	} else if (t1 <= 2.5e+109) {
		tmp = t_1;
	} else {
		tmp = -v / fma(u, 2.0, t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 * v) / Float64(Float64(-Float64(t1 + u)) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -3.5e+136)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= -2.35e-136)
		tmp = t_1;
	elseif (t1 <= 3.6e-186)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif (t1 <= 2.5e+109)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / fma(u, 2.0, t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 * v), $MachinePrecision] / N[((-N[(t1 + u), $MachinePrecision]) * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.5e+136], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, -2.35e-136], t$95$1, If[LessEqual[t1, 3.6e-186], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.5e+109], t$95$1, N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.50000000000000001e136

   1. Initial program 47.7%

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

   3. Taylor expanded in u around 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 93.3

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

   5. Applied rewrites 93.3%

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

   if -3.50000000000000001e136 < t1 < -2.35000000000000011e-136 or 3.5999999999999998e-186 < t1 < 2.5000000000000001e109

   1. Initial program 94.3%

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

   if -2.35000000000000011e-136 < t1 < 3.5999999999999998e-186

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 96.6

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in u around inf

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

      1. lower-/.f64 89.1

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

   10. Applied rewrites 89.1%

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

   if 2.5000000000000001e109 < t1

   1. Initial program 36.3%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 16.0

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

   5. Applied rewrites 16.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 16.4

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

   7. Applied rewrites 16.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 16.4

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

   9. Applied rewrites 16.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 88.2

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

   12. Applied rewrites 88.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -2.35 \cdot 10^{-136}:\\ \;\;\;\;\frac{t1 \cdot v}{\left(-\left(t1 + u\right)\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{-186}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{t1 \cdot v}{\left(-\left(t1 + u\right)\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot t1\\ t_2 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -6 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) (* (+ t1 u) (+ t1 u))) t1))
        (t_2 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -6e+38)
     t_2
     (if (<= t1 -4.5e-146)
       t_1
       (if (<= t1 2.6e-162)
         (* (/ t1 u) (/ (- v) u))
         (if (<= t1 1.6e+135) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / ((t1 + u) * (t1 + u))) * t1;
	double t_2 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -6e+38) {
		tmp = t_2;
	} else if (t1 <= -4.5e-146) {
		tmp = t_1;
	} else if (t1 <= 2.6e-162) {
		tmp = (t1 / u) * (-v / u);
	} else if (t1 <= 1.6e+135) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(t1 + u))) * t1)
	t_2 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -6e+38)
		tmp = t_2;
	elseif (t1 <= -4.5e-146)
		tmp = t_1;
	elseif (t1 <= 2.6e-162)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif (t1 <= 1.6e+135)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6e+38], t$95$2, If[LessEqual[t1, -4.5e-146], t$95$1, If[LessEqual[t1, 2.6e-162], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.6e+135], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -6.0000000000000002e38 or 1.59999999999999987e135 < t1

   1. Initial program 51.3%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 16.4

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

   5. Applied rewrites 16.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 16.7

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

   7. Applied rewrites 16.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 16.7

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

   9. Applied rewrites 16.7%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 90.7

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

   12. Applied rewrites 90.7%

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

   if -6.0000000000000002e38 < t1 < -4.5000000000000001e-146 or 2.6e-162 < t1 < 1.59999999999999987e135

   1. Initial program 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 91.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   if -4.5000000000000001e-146 < t1 < 2.6e-162

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 96.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 87.7

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in u around inf

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

      1. lower-/.f64 89.5

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

   10. Applied rewrites 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{+38}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot t1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -2.3e-122)
     t_1
     (if (<= t1 8.5e-52) (* (/ t1 u) (/ (- v) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -2.3e-122) {
		tmp = t_1;
	} else if (t1 <= 8.5e-52) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -2.3e-122)
		tmp = t_1;
	elseif (t1 <= 8.5e-52)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.3e-122], t$95$1, If[LessEqual[t1, 8.5e-52], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.30000000000000007e-122 or 8.50000000000000006e-52 < t1

   1. Initial program 71.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.2

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

   5. Applied rewrites 29.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 29.0

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

   7. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 28.8

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

   9. Applied rewrites 28.8%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 80.3

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

   12. Applied rewrites 80.3%

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

   if -2.30000000000000007e-122 < t1 < 8.50000000000000006e-52

   1. Initial program 78.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 97.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in u around inf

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

      1. lower-/.f64 85.4

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

   10. Applied rewrites 85.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -2.3e-122) t_1 (if (<= t1 4e-55) (/ (* (- t1) v) (* u u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -2.3e-122) {
		tmp = t_1;
	} else if (t1 <= 4e-55) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -2.3e-122)
		tmp = t_1;
	elseif (t1 <= 4e-55)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.3e-122], t$95$1, If[LessEqual[t1, 4e-55], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.30000000000000007e-122 or 3.99999999999999998e-55 < t1

   1. Initial program 71.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.2

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

   5. Applied rewrites 29.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 29.0

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

   7. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 28.8

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

   9. Applied rewrites 28.8%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 80.3

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

   12. Applied rewrites 80.3%

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

   if -2.30000000000000007e-122 < t1 < 3.99999999999999998e-55

   1. Initial program 78.8%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{t1}{u \cdot u} \cdot \left(-v\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -2.3e-122) t_1 (if (<= t1 4e-55) (* (/ t1 (* u u)) (- v)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -2.3e-122) {
		tmp = t_1;
	} else if (t1 <= 4e-55) {
		tmp = (t1 / (u * u)) * -v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -2.3e-122)
		tmp = t_1;
	elseif (t1 <= 4e-55)
		tmp = Float64(Float64(t1 / Float64(u * u)) * Float64(-v));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.3e-122], t$95$1, If[LessEqual[t1, 4e-55], N[(N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision] * (-v)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.30000000000000007e-122 or 3.99999999999999998e-55 < t1

   1. Initial program 71.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.2

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

   5. Applied rewrites 29.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 29.0

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

   7. Applied rewrites 29.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 28.8

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

   9. Applied rewrites 28.8%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 80.3

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

   12. Applied rewrites 80.3%

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

   if -2.30000000000000007e-122 < t1 < 3.99999999999999998e-55

   1. Initial program 78.8%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{t1}{u \cdot u} \cdot \left(-v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-2neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 99.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 8: 94.7% 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(Float64(-v) / fma(Float64(2.0 + Float64(u / t1)), u, 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 73.9%

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

3. Taylor expanded in u around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 43.2

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

5. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 42.7

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

7. Applied rewrites 42.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. remove-double-neg N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 42.7

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

9. Applied rewrites 42.7%

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

10. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 N/A

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

    6. lower-/.f64 96.9

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

12. Applied rewrites 96.9%

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

13. Final simplification 96.9%

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

Alternative 9: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -4.6 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* 1.0 (/ (- v) u))))
   (if (<= u -4.6e+166) t_1 (if (<= u 2.5e+146) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 1.0 * (-v / u);
	double tmp;
	if (u <= -4.6e+166) {
		tmp = t_1;
	} else if (u <= 2.5e+146) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * (-v / u)
    if (u <= (-4.6d+166)) then
        tmp = t_1
    else if (u <= 2.5d+146) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = 1.0 * (-v / u);
	double tmp;
	if (u <= -4.6e+166) {
		tmp = t_1;
	} else if (u <= 2.5e+146) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 1.0 * (-v / u)
	tmp = 0
	if u <= -4.6e+166:
		tmp = t_1
	elif u <= 2.5e+146:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(1.0 * Float64(Float64(-v) / u))
	tmp = 0.0
	if (u <= -4.6e+166)
		tmp = t_1;
	elseif (u <= 2.5e+146)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 1.0 * (-v / u);
	tmp = 0.0;
	if (u <= -4.6e+166)
		tmp = t_1;
	elseif (u <= 2.5e+146)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(1.0 * N[((-v) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.6e+166], t$95$1, If[LessEqual[u, 2.5e+146], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -4.60000000000000015e166 or 2.4999999999999999e146 < u

   1. Initial program 78.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 98.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 43.2%

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

   if -4.60000000000000015e166 < u < 2.4999999999999999e146

   1. Initial program 72.1%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in u around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 43.2

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

5. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 42.7

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

7. Applied rewrites 42.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. remove-double-neg N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 42.7

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

9. Applied rewrites 42.7%

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

10. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 64.9

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

12. Applied rewrites 64.9%

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

Alternative 11: 61.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / ((-1.0d0) * (t1 + u))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. frac-2neg N/A

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

   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 99.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 64.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 64.4

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 64.4

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

9. Applied rewrites 64.4%

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

Alternative 12: 53.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in u around 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 55.7

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

5. Applied rewrites 55.7%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))