Rosa's DopplerBench

Percentage Accurate: 71.7% → 97.7%

Time: 8.6s

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: 71.7% 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, 0.8× 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 75.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. *-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 98.1

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

   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 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.76 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{v} \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -1.76e+47)
     t_1
     (if (<= t1 5.8e+179) (/ (- t1) (* (/ (+ t1 u) v) (+ t1 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.76e+47) {
		tmp = t_1;
	} else if (t1 <= 5.8e+179) {
		tmp = -t1 / (((t1 + u) / v) * (t1 + u));
	} 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 = -v / (t1 + u)
    if (t1 <= (-1.76d+47)) then
        tmp = t_1
    else if (t1 <= 5.8d+179) then
        tmp = -t1 / (((t1 + u) / v) * (t1 + u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.76e+47) {
		tmp = t_1;
	} else if (t1 <= 5.8e+179) {
		tmp = -t1 / (((t1 + u) / v) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1.76e+47:
		tmp = t_1
	elif t1 <= 5.8e+179:
		tmp = -t1 / (((t1 + u) / v) * (t1 + u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.76e+47)
		tmp = t_1;
	elseif (t1 <= 5.8e+179)
		tmp = Float64(Float64(-t1) / Float64(Float64(Float64(t1 + u) / v) * Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1.76e+47)
		tmp = t_1;
	elseif (t1 <= 5.8e+179)
		tmp = -t1 / (((t1 + u) / v) * (t1 + u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.76e+47], t$95$1, If[LessEqual[t1, 5.8e+179], N[((-t1) / N[(N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.76e47 or 5.80000000000000038e179 < t1

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   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 91.2

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

   7. Applied rewrites 91.2%

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

   if -1.76e47 < t1 < 5.80000000000000038e179

   1. Initial program 84.2%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. frac-2neg N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 92.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 92.1

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

   4. Applied rewrites 92.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.76 \cdot 10^{+47}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{v} \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.9 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -3.9e+156)
     t_1
     (if (<= t1 1.6e+76) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.9e+156) {
		tmp = t_1;
	} else if (t1 <= 1.6e+76) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} 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 = -v / (t1 + u)
    if (t1 <= (-3.9d+156)) then
        tmp = t_1
    else if (t1 <= 1.6d+76) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.9e+156) {
		tmp = t_1;
	} else if (t1 <= 1.6e+76) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -3.9e+156:
		tmp = t_1
	elif t1 <= 1.6e+76:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -3.9e+156)
		tmp = t_1;
	elseif (t1 <= 1.6e+76)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -3.9e+156)
		tmp = t_1;
	elseif (t1 <= 1.6e+76)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.9e+156], t$95$1, If[LessEqual[t1, 1.6e+76], 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 < -3.8999999999999997e156 or 1.59999999999999988e76 < t1

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.9

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

   7. Applied rewrites 88.9%

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

   if -3.8999999999999997e156 < t1 < 1.59999999999999988e76

   1. Initial program 86.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{+156}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -7.4e-44) t_1 (if (<= t1 8.2e-7) (/ (* (/ v u) t1) (- u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.4e-44) {
		tmp = t_1;
	} else if (t1 <= 8.2e-7) {
		tmp = ((v / u) * t1) / -u;
	} 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 = -v / (t1 + u)
    if (t1 <= (-7.4d-44)) then
        tmp = t_1
    else if (t1 <= 8.2d-7) then
        tmp = ((v / u) * t1) / -u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.4e-44) {
		tmp = t_1;
	} else if (t1 <= 8.2e-7) {
		tmp = ((v / u) * t1) / -u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -7.4e-44:
		tmp = t_1
	elif t1 <= 8.2e-7:
		tmp = ((v / u) * t1) / -u
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -7.4e-44)
		tmp = t_1;
	elseif (t1 <= 8.2e-7)
		tmp = Float64(Float64(Float64(v / u) * t1) / Float64(-u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -7.4e-44)
		tmp = t_1;
	elseif (t1 <= 8.2e-7)
		tmp = ((v / u) * t1) / -u;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.4e-44], t$95$1, If[LessEqual[t1, 8.2e-7], N[(N[(N[(v / u), $MachinePrecision] * t1), $MachinePrecision] / (-u)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.4e-44 or 8.1999999999999998e-7 < t1

   1. Initial program 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in 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 83.0

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

   7. Applied rewrites 83.0%

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

   if -7.4e-44 < t1 < 8.1999999999999998e-7

   1. Initial program 86.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 96.1

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

      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 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in u around inf

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

      1. lower-/.f64 84.4

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

   10. Applied rewrites 84.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -1e-43) t_1 (if (<= t1 8.2e-7) (/ (/ (* t1 v) u) (- u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1e-43) {
		tmp = t_1;
	} else if (t1 <= 8.2e-7) {
		tmp = ((t1 * v) / u) / -u;
	} 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 = -v / (t1 + u)
    if (t1 <= (-1d-43)) then
        tmp = t_1
    else if (t1 <= 8.2d-7) then
        tmp = ((t1 * v) / u) / -u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1e-43) {
		tmp = t_1;
	} else if (t1 <= 8.2e-7) {
		tmp = ((t1 * v) / u) / -u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1e-43:
		tmp = t_1
	elif t1 <= 8.2e-7:
		tmp = ((t1 * v) / u) / -u
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1e-43)
		tmp = t_1;
	elseif (t1 <= 8.2e-7)
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(-u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1e-43)
		tmp = t_1;
	elseif (t1 <= 8.2e-7)
		tmp = ((t1 * v) / u) / -u;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1e-43], t$95$1, If[LessEqual[t1, 8.2e-7], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / (-u)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000008e-43 or 8.1999999999999998e-7 < t1

   1. Initial program 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in 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 83.0

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

   7. Applied rewrites 83.0%

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

   if -1.00000000000000008e-43 < t1 < 8.1999999999999998e-7

   1. Initial program 86.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 96.1

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

      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 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 81.6

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

   10. Applied rewrites 81.6%

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

   12. Applied rewrites 83.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-43}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 0.0085:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -1e-43) t_1 (if (<= t1 0.0085) (/ t1 (* (/ u v) (- u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1e-43) {
		tmp = t_1;
	} else if (t1 <= 0.0085) {
		tmp = t1 / ((u / v) * -u);
	} 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 = -v / (t1 + u)
    if (t1 <= (-1d-43)) then
        tmp = t_1
    else if (t1 <= 0.0085d0) then
        tmp = t1 / ((u / v) * -u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1e-43) {
		tmp = t_1;
	} else if (t1 <= 0.0085) {
		tmp = t1 / ((u / v) * -u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1e-43:
		tmp = t_1
	elif t1 <= 0.0085:
		tmp = t1 / ((u / v) * -u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1e-43)
		tmp = t_1;
	elseif (t1 <= 0.0085)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(-u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1e-43)
		tmp = t_1;
	elseif (t1 <= 0.0085)
		tmp = t1 / ((u / v) * -u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1e-43], t$95$1, If[LessEqual[t1, 0.0085], N[(t1 / N[(N[(u / v), $MachinePrecision] * (-u)), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000008e-43 or 0.0085000000000000006 < t1

   1. Initial program 64.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -1.00000000000000008e-43 < t1 < 0.0085000000000000006

   1. Initial program 86.6%

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

      1. lift-/.f64 N/A

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

      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 96.1

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

      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 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. metadata-eval N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 80.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 80.3

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

   9. Applied rewrites 80.3%

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

   10. Taylor expanded in u around inf

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

       1. lower-/.f64 82.1

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

   12. Applied rewrites 82.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-43}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 0.0085:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -1e-43) t_1 (if (<= t1 8.2e-7) (* (/ (- v) u) (/ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1e-43) {
		tmp = t_1;
	} else if (t1 <= 8.2e-7) {
		tmp = (-v / u) * (t1 / u);
	} 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 = -v / (t1 + u)
    if (t1 <= (-1d-43)) then
        tmp = t_1
    else if (t1 <= 8.2d-7) then
        tmp = (-v / u) * (t1 / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1e-43) {
		tmp = t_1;
	} else if (t1 <= 8.2e-7) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1e-43:
		tmp = t_1
	elif t1 <= 8.2e-7:
		tmp = (-v / u) * (t1 / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1e-43)
		tmp = t_1;
	elseif (t1 <= 8.2e-7)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1e-43)
		tmp = t_1;
	elseif (t1 <= 8.2e-7)
		tmp = (-v / u) * (t1 / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1e-43], t$95$1, If[LessEqual[t1, 8.2e-7], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000008e-43 or 8.1999999999999998e-7 < t1

   1. Initial program 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in 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 83.0

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

   7. Applied rewrites 83.0%

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

   if -1.00000000000000008e-43 < t1 < 8.1999999999999998e-7

   1. Initial program 86.4%

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

   3. Taylor expanded in u around 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 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-43}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 0.0042:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -7.4e-44) t_1 (if (<= t1 0.0042) (* (/ (- v) (* u u)) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.4e-44) {
		tmp = t_1;
	} else if (t1 <= 0.0042) {
		tmp = (-v / (u * u)) * 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 = -v / (t1 + u)
    if (t1 <= (-7.4d-44)) then
        tmp = t_1
    else if (t1 <= 0.0042d0) then
        tmp = (-v / (u * u)) * 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 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.4e-44) {
		tmp = t_1;
	} else if (t1 <= 0.0042) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -7.4e-44:
		tmp = t_1
	elif t1 <= 0.0042:
		tmp = (-v / (u * u)) * t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -7.4e-44)
		tmp = t_1;
	elseif (t1 <= 0.0042)
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -7.4e-44)
		tmp = t_1;
	elseif (t1 <= 0.0042)
		tmp = (-v / (u * u)) * t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.4e-44], t$95$1, If[LessEqual[t1, 0.0042], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.4e-44 or 0.00419999999999999974 < t1

   1. Initial program 64.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -7.4e-44 < t1 < 0.00419999999999999974

   1. Initial program 86.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 0.0042:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 0.0042:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -3.4e-45) t_1 (if (<= t1 0.0042) (* (/ (- t1) (* u u)) v) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.4e-45) {
		tmp = t_1;
	} else if (t1 <= 0.0042) {
		tmp = (-t1 / (u * u)) * v;
	} 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 = -v / (t1 + u)
    if (t1 <= (-3.4d-45)) then
        tmp = t_1
    else if (t1 <= 0.0042d0) then
        tmp = (-t1 / (u * u)) * v
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.4e-45) {
		tmp = t_1;
	} else if (t1 <= 0.0042) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -3.4e-45:
		tmp = t_1
	elif t1 <= 0.0042:
		tmp = (-t1 / (u * u)) * v
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -3.4e-45)
		tmp = t_1;
	elseif (t1 <= 0.0042)
		tmp = Float64(Float64(Float64(-t1) / Float64(u * u)) * v);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -3.4e-45)
		tmp = t_1;
	elseif (t1 <= 0.0042)
		tmp = (-t1 / (u * u)) * v;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e-45], t$95$1, If[LessEqual[t1, 0.0042], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.40000000000000004e-45 or 0.00419999999999999974 < t1

   1. Initial program 64.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -3.40000000000000004e-45 < t1 < 0.00419999999999999974

   1. Initial program 86.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.7

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

   7. Applied rewrites 71.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 0.0042:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.0% 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 75.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 96.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 96.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 96.8

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

4. Applied rewrites 96.8%

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

5. Final simplification 96.8%

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

Alternative 11: 62.0% 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 75.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 96.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 96.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 96.8

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

4. Applied rewrites 96.8%

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

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 57.2

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

7. Applied rewrites 57.2%

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

8. Final simplification 57.2%

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

Alternative 12: 54.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 50.8

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

5. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))