Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.1%

Time: 7.1s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. lift-/.f64 N/A

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

   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. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 97.8

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 97.8

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

4. Applied rewrites 97.8%

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

5. Final simplification 97.8%

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -2.05 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -2.05e+97)
     t_1
     (if (<= t1 1.6e+58) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -2.05e+97) {
		tmp = t_1;
	} else if (t1 <= 1.6e+58) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -2.05e+97)
		tmp = t_1;
	elseif (t1 <= 1.6e+58)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.05e+97], t$95$1, If[LessEqual[t1, 1.6e+58], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.04999999999999994e97 or 1.60000000000000008e58 < t1

   1. Initial program 47.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 95.5

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

      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 95.5

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

      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 95.5

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

   4. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 95.5

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

   6. Applied rewrites 95.5%

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

   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 92.0

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

   9. Applied rewrites 92.0%

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

   if -2.04999999999999994e97 < t1 < 1.60000000000000008e58

   1. Initial program 83.4%

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

Alternative 3: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -3.4e+22)
     t_1
     (if (<= t1 2.6e+54) (/ (* (/ (- v) u) t1) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -3.4e+22) {
		tmp = t_1;
	} else if (t1 <= 2.6e+54) {
		tmp = ((-v / u) * t1) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -3.4e+22)
		tmp = t_1;
	elseif (t1 <= 2.6e+54)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e+22], t$95$1, If[LessEqual[t1, 2.6e+54], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e22 or 2.60000000000000007e54 < t1

   1. Initial program 51.1%

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

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

      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 95.3

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

      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 95.3

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

      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 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 95.3

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

   6. Applied rewrites 95.3%

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

   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 91.8

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

   9. Applied rewrites 91.8%

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

   if -3.4e22 < t1 < 2.60000000000000007e54

   1. Initial program 82.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 75.8%

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

Alternative 4: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -3.4e+22)
     t_1
     (if (<= t1 2.6e+54) (/ (* (/ t1 u) v) (- u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -3.4e+22) {
		tmp = t_1;
	} else if (t1 <= 2.6e+54) {
		tmp = ((t1 / u) * v) / -u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -3.4e+22)
		tmp = t_1;
	elseif (t1 <= 2.6e+54)
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e+22], t$95$1, If[LessEqual[t1, 2.6e+54], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e22 or 2.60000000000000007e54 < t1

   1. Initial program 51.1%

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

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

      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 95.3

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

      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 95.3

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

      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 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 95.3

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

   6. Applied rewrites 95.3%

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

   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 91.8

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

   9. Applied rewrites 91.8%

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

   if -3.4e22 < t1 < 2.60000000000000007e54

   1. Initial program 82.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -3.4e+22)
     t_1
     (if (<= t1 2.6e+54) (* (/ t1 u) (/ (- v) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -3.4e+22) {
		tmp = t_1;
	} else if (t1 <= 2.6e+54) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -3.4e+22)
		tmp = t_1;
	elseif (t1 <= 2.6e+54)
		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[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e+22], t$95$1, If[LessEqual[t1, 2.6e+54], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e22 or 2.60000000000000007e54 < t1

   1. Initial program 51.1%

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

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

      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 95.3

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

      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 95.3

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

      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 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 95.3

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

   6. Applied rewrites 95.3%

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

   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 91.8

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

   9. Applied rewrites 91.8%

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

   if -3.4e22 < t1 < 2.60000000000000007e54

   1. Initial program 82.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{-v}{u}}{u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -3.4e+22)
     t_1
     (if (<= t1 2.6e+54) (* (/ (/ (- v) u) u) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -3.4e+22) {
		tmp = t_1;
	} else if (t1 <= 2.6e+54) {
		tmp = ((-v / u) / u) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -3.4e+22)
		tmp = t_1;
	elseif (t1 <= 2.6e+54)
		tmp = Float64(Float64(Float64(Float64(-v) / u) / u) * t1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e+22], t$95$1, If[LessEqual[t1, 2.6e+54], N[(N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e22 or 2.60000000000000007e54 < t1

   1. Initial program 51.1%

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

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

      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 95.3

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

      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 95.3

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

      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 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 95.3

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

   6. Applied rewrites 95.3%

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

   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 91.8

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

   9. Applied rewrites 91.8%

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

   if -3.4e22 < t1 < 2.60000000000000007e54

   1. Initial program 82.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 72.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{-v}{u}}{u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 6.6 \cdot 10^{+211}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2 - \frac{u}{t1}, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot \left(-t1\right)}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 6.6e+211)
   (/ v (fma (- -2.0 (/ u t1)) u (- t1)))
   (/ (* (/ v u) (- t1)) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 6.6e+211) {
		tmp = v / fma((-2.0 - (u / t1)), u, -t1);
	} else {
		tmp = ((v / u) * -t1) / (t1 + u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 6.6e+211)
		tmp = Float64(v / fma(Float64(-2.0 - Float64(u / t1)), u, Float64(-t1)));
	else
		tmp = Float64(Float64(Float64(v / u) * Float64(-t1)) / Float64(t1 + u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 6.6e+211], N[(v / N[(N[(-2.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / u), $MachinePrecision] * (-t1)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 6.59999999999999966e211

   1. Initial program 68.9%

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

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

      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 96.2

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

      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 96.2

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

      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 96.2

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

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 96.2

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

   6. Applied rewrites 96.2%

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

   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 96.2

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

   9. Applied rewrites 96.2%

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

   if 6.59999999999999966e211 < u

   1. Initial program 63.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-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. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 95.3

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

   7. Applied rewrites 95.3%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 6.6 \cdot 10^{+211}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2 - \frac{u}{t1}, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot \left(-t1\right)}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -6.6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -6.6e+21)
     t_1
     (if (<= t1 2.6e+54) (* (/ v (* (- u) u)) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / fma(-2.0, u, -t1);
	double tmp;
	if (t1 <= -6.6e+21) {
		tmp = t_1;
	} else if (t1 <= 2.6e+54) {
		tmp = (v / (-u * u)) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -6.6e+21)
		tmp = t_1;
	elseif (t1 <= 2.6e+54)
		tmp = Float64(Float64(v / Float64(Float64(-u) * u)) * t1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6.6e+21], t$95$1, If[LessEqual[t1, 2.6e+54], N[(N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.6e21 or 2.60000000000000007e54 < t1

   1. Initial program 51.1%

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

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

      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 95.3

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

      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 95.3

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

      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 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
   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 95.3

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

   6. Applied rewrites 95.3%

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

   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 91.8

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

   9. Applied rewrites 91.8%

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

   if -6.6e21 < t1 < 2.60000000000000007e54

   1. Initial program 82.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 69.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 0.8× speedup

Math

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

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

Derivation

1. Initial program 68.4%

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

   1. lift-/.f64 N/A

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

   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 97.6

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

   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 97.6

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

   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 97.6

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

4. Applied rewrites 97.6%

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

5. Final simplification 97.6%

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

Alternative 10: 61.4% 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 68.4%

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

   1. lift-/.f64 N/A

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

   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 94.5

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

   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 94.5

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

   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 94.5

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

4. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{-\left(u + t1\right)}{t1} \cdot \left(u + t1\right)}} \]
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 94.5

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

6. Applied rewrites 94.5%

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

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 62.9

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

9. Applied rewrites 62.9%

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

Alternative 11: 60.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. lift-/.f64 N/A

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

   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 97.6

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

   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 97.6

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

   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 97.6

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 62.3

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

7. Applied rewrites 62.3%

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

8. Final simplification 62.3%

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

Alternative 12: 53.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 56.4

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

5. Applied rewrites 56.4%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))