Rosa's DopplerBench

Percentage Accurate: 73.6% → 98.0%

Time: 7.8s

Alternatives: 14

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: 73.6% 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: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

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

   4. times-frac N/A

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

   5. *-commutative 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 98.4

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

   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 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{if}\;t1 \leq -4.8 \cdot 10^{+75}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -9.4 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
        (t_2 (/ (* v 1.0) (- (+ u t1)))))
   (if (<= t1 -4.8e+75)
     t_2
     (if (<= t1 -9.4e-145)
       t_1
       (if (<= t1 6.5e-114)
         (/ t1 (* (- u) (/ u v)))
         (if (<= t1 2.4e+109) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = (v * 1.0) / -(u + t1);
	double tmp;
	if (t1 <= -4.8e+75) {
		tmp = t_2;
	} else if (t1 <= -9.4e-145) {
		tmp = t_1;
	} else if (t1 <= 6.5e-114) {
		tmp = t1 / (-u * (u / v));
	} else if (t1 <= 2.4e+109) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
    t_2 = (v * 1.0d0) / -(u + t1)
    if (t1 <= (-4.8d+75)) then
        tmp = t_2
    else if (t1 <= (-9.4d-145)) then
        tmp = t_1
    else if (t1 <= 6.5d-114) then
        tmp = t1 / (-u * (u / v))
    else if (t1 <= 2.4d+109) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = (v * 1.0) / -(u + t1);
	double tmp;
	if (t1 <= -4.8e+75) {
		tmp = t_2;
	} else if (t1 <= -9.4e-145) {
		tmp = t_1;
	} else if (t1 <= 6.5e-114) {
		tmp = t1 / (-u * (u / v));
	} else if (t1 <= 2.4e+109) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
	t_2 = (v * 1.0) / -(u + t1)
	tmp = 0
	if t1 <= -4.8e+75:
		tmp = t_2
	elif t1 <= -9.4e-145:
		tmp = t_1
	elif t1 <= 6.5e-114:
		tmp = t1 / (-u * (u / v))
	elif t1 <= 2.4e+109:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	t_2 = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)))
	tmp = 0.0
	if (t1 <= -4.8e+75)
		tmp = t_2;
	elseif (t1 <= -9.4e-145)
		tmp = t_1;
	elseif (t1 <= 6.5e-114)
		tmp = Float64(t1 / Float64(Float64(-u) * Float64(u / v)));
	elseif (t1 <= 2.4e+109)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	t_2 = (v * 1.0) / -(u + t1);
	tmp = 0.0;
	if (t1 <= -4.8e+75)
		tmp = t_2;
	elseif (t1 <= -9.4e-145)
		tmp = t_1;
	elseif (t1 <= 6.5e-114)
		tmp = t1 / (-u * (u / v));
	elseif (t1 <= 2.4e+109)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t1, -4.8e+75], t$95$2, If[LessEqual[t1, -9.4e-145], t$95$1, If[LessEqual[t1, 6.5e-114], N[(t1 / N[((-u) * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.4e+109], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.8e75 or 2.39999999999999987e109 < t1

   1. Initial program 42.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.5

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 91.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 91.7

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

   9. Applied rewrites 91.7%

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

   if -4.8e75 < t1 < -9.4000000000000004e-145 or 6.4999999999999998e-114 < t1 < 2.39999999999999987e109

   1. Initial program 95.6%

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

   if -9.4000000000000004e-145 < t1 < 6.4999999999999998e-114

   1. Initial program 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      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 92.9

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

      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 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 88.9

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

   7. Applied rewrites 88.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{elif}\;t1 \leq -9.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e-123) (not (<= t1 1450.0)))
   (/ (* v 1.0) (- (+ u t1)))
   (/ t1 (* (- u) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = t1 / (-u * (u / v));
	}
	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) :: tmp
    if ((t1 <= (-1.65d-123)) .or. (.not. (t1 <= 1450.0d0))) then
        tmp = (v * 1.0d0) / -(u + t1)
    else
        tmp = t1 / (-u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = t1 / (-u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e-123) or not (t1 <= 1450.0):
		tmp = (v * 1.0) / -(u + t1)
	else:
		tmp = t1 / (-u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0))
		tmp = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)));
	else
		tmp = Float64(t1 / Float64(Float64(-u) * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.65e-123) || ~((t1 <= 1450.0)))
		tmp = (v * 1.0) / -(u + t1);
	else
		tmp = t1 / (-u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e-123], N[Not[LessEqual[t1, 1450.0]], $MachinePrecision]], N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision], N[(t1 / N[((-u) * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e-123 or 1450 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

   if -1.6500000000000001e-123 < t1 < 1450

   1. Initial program 81.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 92.7

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

      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 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 86.1

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

   7. Applied rewrites 86.1%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{u} \cdot \left(-t1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e-123) (not (<= t1 1450.0)))
   (/ (* v 1.0) (- (+ u t1)))
   (* (/ (/ v u) u) (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = ((v / u) / u) * -t1;
	}
	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) :: tmp
    if ((t1 <= (-1.65d-123)) .or. (.not. (t1 <= 1450.0d0))) then
        tmp = (v * 1.0d0) / -(u + t1)
    else
        tmp = ((v / u) / u) * -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = ((v / u) / u) * -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e-123) or not (t1 <= 1450.0):
		tmp = (v * 1.0) / -(u + t1)
	else:
		tmp = ((v / u) / u) * -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0))
		tmp = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)));
	else
		tmp = Float64(Float64(Float64(v / u) / u) * Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.65e-123) || ~((t1 <= 1450.0)))
		tmp = (v * 1.0) / -(u + t1);
	else
		tmp = ((v / u) / u) * -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e-123], N[Not[LessEqual[t1, 1450.0]], $MachinePrecision]], N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision], N[(N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision] * (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e-123 or 1450 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

   if -1.6500000000000001e-123 < t1 < 1450

   1. Initial program 81.1%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 81.0

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 86.0

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

   10. Applied rewrites 86.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{u} \cdot \left(-t1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e-123) (not (<= t1 1450.0)))
   (/ (* v 1.0) (- (+ u t1)))
   (/ (* (/ v u) t1) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = ((v / u) * t1) / -u;
	}
	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) :: tmp
    if ((t1 <= (-1.65d-123)) .or. (.not. (t1 <= 1450.0d0))) then
        tmp = (v * 1.0d0) / -(u + t1)
    else
        tmp = ((v / u) * t1) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = ((v / u) * t1) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e-123) or not (t1 <= 1450.0):
		tmp = (v * 1.0) / -(u + t1)
	else:
		tmp = ((v / u) * t1) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0))
		tmp = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)));
	else
		tmp = Float64(Float64(Float64(v / u) * t1) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.65e-123) || ~((t1 <= 1450.0)))
		tmp = (v * 1.0) / -(u + t1);
	else
		tmp = ((v / u) * t1) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e-123], N[Not[LessEqual[t1, 1450.0]], $MachinePrecision]], N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision], N[(N[(N[(v / u), $MachinePrecision] * t1), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e-123 or 1450 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

   if -1.6500000000000001e-123 < t1 < 1450

   1. Initial program 81.1%

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

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

      4. times-frac N/A

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

      5. *-commutative 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 95.6

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

      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 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e-123) (not (<= t1 1450.0)))
   (/ (* v 1.0) (- (+ u t1)))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	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) :: tmp
    if ((t1 <= (-1.65d-123)) .or. (.not. (t1 <= 1450.0d0))) then
        tmp = (v * 1.0d0) / -(u + t1)
    else
        tmp = (t1 / u) * (-v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e-123) or not (t1 <= 1450.0):
		tmp = (v * 1.0) / -(u + t1)
	else:
		tmp = (t1 / u) * (-v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0))
		tmp = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.65e-123) || ~((t1 <= 1450.0)))
		tmp = (v * 1.0) / -(u + t1);
	else
		tmp = (t1 / u) * (-v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e-123], N[Not[LessEqual[t1, 1450.0]], $MachinePrecision]], N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e-123 or 1450 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

   if -1.6500000000000001e-123 < t1 < 1450

   1. Initial program 81.1%

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

   3. Taylor expanded in u around 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 84.6

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

   5. Applied rewrites 84.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 3.25 \cdot 10^{+233}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{t1 + u} \cdot \left(-v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 3.25e+233)
   (* (/ (/ t1 (+ t1 u)) (+ t1 u)) (- v))
   (/ (- t1) (* (/ (+ u t1) v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 3.25e+233) {
		tmp = ((t1 / (t1 + u)) / (t1 + u)) * -v;
	} else {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	}
	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) :: tmp
    if (v <= 3.25d+233) then
        tmp = ((t1 / (t1 + u)) / (t1 + u)) * -v
    else
        tmp = -t1 / (((u + t1) / v) * (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 3.25e+233) {
		tmp = ((t1 / (t1 + u)) / (t1 + u)) * -v;
	} else {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 3.25e+233:
		tmp = ((t1 / (t1 + u)) / (t1 + u)) * -v
	else:
		tmp = -t1 / (((u + t1) / v) * (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 3.25e+233)
		tmp = Float64(Float64(Float64(t1 / Float64(t1 + u)) / Float64(t1 + u)) * Float64(-v));
	else
		tmp = Float64(Float64(-t1) / Float64(Float64(Float64(u + t1) / v) * Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 3.25e+233)
		tmp = ((t1 / (t1 + u)) / (t1 + u)) * -v;
	else
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 3.25e+233], N[(N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * (-v)), $MachinePrecision], N[((-t1) / N[(N[(N[(u + t1), $MachinePrecision] / v), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.25000000000000019e233

   1. Initial program 71.3%

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

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

      4. times-frac N/A

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

      5. *-commutative 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 98.2

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

      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 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. lift-neg.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-/.f64 N/A

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

      17. frac-2neg N/A

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

      18. metadata-eval N/A

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

      19. lift-neg.f64 N/A

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

      20. div-inv N/A

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

   6. Applied rewrites 96.7%

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

   if 3.25000000000000019e233 < v

   1. Initial program 44.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 3.25 \cdot 10^{+233}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{t1 + u} \cdot \left(-v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e-123) (not (<= t1 1450.0)))
   (/ (* v 1.0) (- (+ u t1)))
   (* (/ (- v) (* u u)) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = (-v / (u * u)) * t1;
	}
	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) :: tmp
    if ((t1 <= (-1.65d-123)) .or. (.not. (t1 <= 1450.0d0))) then
        tmp = (v * 1.0d0) / -(u + t1)
    else
        tmp = (-v / (u * u)) * t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = (-v / (u * u)) * t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e-123) or not (t1 <= 1450.0):
		tmp = (v * 1.0) / -(u + t1)
	else:
		tmp = (-v / (u * u)) * t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0))
		tmp = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)));
	else
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.65e-123) || ~((t1 <= 1450.0)))
		tmp = (v * 1.0) / -(u + t1);
	else
		tmp = (-v / (u * u)) * t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e-123], N[Not[LessEqual[t1, 1450.0]], $MachinePrecision]], N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e-123 or 1450 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

   if -1.6500000000000001e-123 < t1 < 1450

   1. Initial program 81.1%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 81.0

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

   7. Applied rewrites 81.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e-123) (not (<= t1 1450.0)))
   (/ (* v 1.0) (- (+ u t1)))
   (* v (/ (- t1) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	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) :: tmp
    if ((t1 <= (-1.65d-123)) .or. (.not. (t1 <= 1450.0d0))) then
        tmp = (v * 1.0d0) / -(u + t1)
    else
        tmp = v * (-t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0)) {
		tmp = (v * 1.0) / -(u + t1);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e-123) or not (t1 <= 1450.0):
		tmp = (v * 1.0) / -(u + t1)
	else:
		tmp = v * (-t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e-123) || !(t1 <= 1450.0))
		tmp = Float64(Float64(v * 1.0) / Float64(-Float64(u + t1)));
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.65e-123) || ~((t1 <= 1450.0)))
		tmp = (v * 1.0) / -(u + t1);
	else
		tmp = v * (-t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e-123], N[Not[LessEqual[t1, 1450.0]], $MachinePrecision]], N[(N[(v * 1.0), $MachinePrecision] / (-N[(u + t1), $MachinePrecision])), $MachinePrecision], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e-123 or 1450 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

   if -1.6500000000000001e-123 < t1 < 1450

   1. Initial program 81.1%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 72.9

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

   7. Applied rewrites 72.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-123} \lor \neg \left(t1 \leq 1450\right):\\ \;\;\;\;\frac{v \cdot 1}{-\left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

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

Alternative 11: 95.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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)) / (t1 + u)) * -v
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

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

   4. times-frac N/A

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

   5. *-commutative 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 98.4

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

   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 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. lift-neg.f64 N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r/ N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lift-/.f64 N/A

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

   17. frac-2neg N/A

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

   18. metadata-eval N/A

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

   19. lift-neg.f64 N/A

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

   20. div-inv N/A

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

6. Applied rewrites 95.8%

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

7. Final simplification 95.8%

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

Alternative 12: 61.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. neg-mul-1 N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. distribute-lft-neg-out N/A

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

   16. remove-double-neg N/A

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

   17. *-commutative N/A

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

   18. associate-/l* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-/.f64 97.4

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 97.4

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 65.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 66.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 66.0

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

9. Applied rewrites 66.0%

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

10. Final simplification 66.0%

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

Alternative 13: 61.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

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

   4. times-frac N/A

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

   5. *-commutative 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 98.4

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

   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 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. lift-neg.f64 N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r/ N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lift-/.f64 N/A

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

   17. frac-2neg N/A

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

   18. metadata-eval N/A

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

   19. lift-neg.f64 N/A

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

   20. div-inv N/A

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

6. Applied rewrites 95.8%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 65.8%

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

Alternative 14: 54.2% 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 69.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 59.8

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

5. Applied rewrites 59.8%

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

6. Final simplification 59.8%

   \[\leadsto \frac{-v}{t1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024317 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))