Rosa's DopplerBench

Percentage Accurate: 73.1% → 97.8%

Time: 7.3s

Alternatives: 9

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.2

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

4. Applied rewrites 98.2%

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

5. Final simplification 98.2%

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

Alternative 2: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2.25 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-249}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{elif}\;t1 \leq 1.66 \cdot 10^{+55}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2.25e+61)
     t_1
     (if (<= t1 3.8e-249)
       (/ (* (- t1) v) (* (+ u t1) (+ u t1)))
       (if (<= t1 1.66e+55) (/ (- t1) (* (/ (+ u t1) v) (+ u t1))) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2.25e+61) {
		tmp = t_1;
	} else if (t1 <= 3.8e-249) {
		tmp = (-t1 * v) / ((u + t1) * (u + t1));
	} else if (t1 <= 1.66e+55) {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-2.25d+61)) then
        tmp = t_1
    else if (t1 <= 3.8d-249) then
        tmp = (-t1 * v) / ((u + t1) * (u + t1))
    else if (t1 <= 1.66d+55) then
        tmp = -t1 / (((u + t1) / v) * (u + t1))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2.25e+61) {
		tmp = t_1;
	} else if (t1 <= 3.8e-249) {
		tmp = (-t1 * v) / ((u + t1) * (u + t1));
	} else if (t1 <= 1.66e+55) {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -2.25e+61:
		tmp = t_1
	elif t1 <= 3.8e-249:
		tmp = (-t1 * v) / ((u + t1) * (u + t1))
	elif t1 <= 1.66e+55:
		tmp = -t1 / (((u + t1) / v) * (u + t1))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -2.25e+61)
		tmp = t_1;
	elseif (t1 <= 3.8e-249)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(u + t1) * Float64(u + t1)));
	elseif (t1 <= 1.66e+55)
		tmp = Float64(Float64(-t1) / Float64(Float64(Float64(u + t1) / v) * Float64(u + t1)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -2.25e+61)
		tmp = t_1;
	elseif (t1 <= 3.8e-249)
		tmp = (-t1 * v) / ((u + t1) * (u + t1));
	elseif (t1 <= 1.66e+55)
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.25e+61], t$95$1, If[LessEqual[t1, 3.8e-249], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.66e+55], N[((-t1) / N[(N[(N[(u + t1), $MachinePrecision] / v), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.25e61 or 1.6599999999999999e55 < t1

   1. Initial program 47.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.7

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

   7. Applied rewrites 86.7%

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

   if -2.25e61 < t1 < 3.8000000000000001e-249

   1. Initial program 93.3%

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

   if 3.8000000000000001e-249 < t1 < 1.6599999999999999e55

   1. Initial program 89.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 96.6

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

      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 96.6

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

   4. Applied rewrites 96.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-249}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{elif}\;t1 \leq 1.66 \cdot 10^{+55}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2.25 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2.25e+61)
     t_1
     (if (<= t1 1.6e+55) (/ (* (- t1) v) (* (+ u t1) (+ u t1))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2.25e+61) {
		tmp = t_1;
	} else if (t1 <= 1.6e+55) {
		tmp = (-t1 * v) / ((u + t1) * (u + t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-2.25d+61)) then
        tmp = t_1
    else if (t1 <= 1.6d+55) then
        tmp = (-t1 * v) / ((u + t1) * (u + t1))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2.25e+61) {
		tmp = t_1;
	} else if (t1 <= 1.6e+55) {
		tmp = (-t1 * v) / ((u + t1) * (u + t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -2.25e+61:
		tmp = t_1
	elif t1 <= 1.6e+55:
		tmp = (-t1 * v) / ((u + t1) * (u + t1))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -2.25e+61)
		tmp = t_1;
	elseif (t1 <= 1.6e+55)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(u + t1) * Float64(u + t1)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -2.25e+61)
		tmp = t_1;
	elseif (t1 <= 1.6e+55)
		tmp = (-t1 * v) / ((u + t1) * (u + t1));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.25e+61], t$95$1, If[LessEqual[t1, 1.6e+55], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.25e61 or 1.6000000000000001e55 < t1

   1. Initial program 47.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.7

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

   7. Applied rewrites 86.7%

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

   if -2.25e61 < t1 < 1.6000000000000001e55

   1. Initial program 91.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -1.8e-128)
     t_1
     (if (<= t1 2.9e-74) (* (/ (- v) u) (/ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -1.8e-128) {
		tmp = t_1;
	} else if (t1 <= 2.9e-74) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-1.8d-128)) then
        tmp = t_1
    else if (t1 <= 2.9d-74) then
        tmp = (-v / u) * (t1 / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -1.8e-128) {
		tmp = t_1;
	} else if (t1 <= 2.9e-74) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -1.8e-128:
		tmp = t_1
	elif t1 <= 2.9e-74:
		tmp = (-v / u) * (t1 / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -1.8e-128)
		tmp = t_1;
	elseif (t1 <= 2.9e-74)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -1.8e-128)
		tmp = t_1;
	elseif (t1 <= 2.9e-74)
		tmp = (-v / u) * (t1 / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e-128], t$95$1, If[LessEqual[t1, 2.9e-74], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.80000000000000012e-128 or 2.9e-74 < t1

   1. Initial program 63.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if -1.80000000000000012e-128 < t1 < 2.9e-74

   1. Initial program 87.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -1.8e-128)
     t_1
     (if (<= t1 2.9e-74) (* (/ (- t1) (* u u)) v) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -1.8e-128) {
		tmp = t_1;
	} else if (t1 <= 2.9e-74) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-1.8d-128)) then
        tmp = t_1
    else if (t1 <= 2.9d-74) then
        tmp = (-t1 / (u * u)) * v
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -1.8e-128) {
		tmp = t_1;
	} else if (t1 <= 2.9e-74) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -1.8e-128:
		tmp = t_1
	elif t1 <= 2.9e-74:
		tmp = (-t1 / (u * u)) * v
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -1.8e-128)
		tmp = t_1;
	elseif (t1 <= 2.9e-74)
		tmp = Float64(Float64(Float64(-t1) / Float64(u * u)) * v);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -1.8e-128)
		tmp = t_1;
	elseif (t1 <= 2.9e-74)
		tmp = (-t1 / (u * u)) * v;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e-128], t$95$1, If[LessEqual[t1, 2.9e-74], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.80000000000000012e-128 or 2.9e-74 < t1

   1. Initial program 63.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if -1.80000000000000012e-128 < t1 < 2.9e-74

   1. Initial program 87.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 85.7

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

   5. Applied rewrites 85.7%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
   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 87.2

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

   7. Applied rewrites 87.2%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{u \cdot u} \cdot v\\ \mathbf{if}\;u \leq -1.3 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 (* u u)) v)))
   (if (<= u -1.3e+128) t_1 (if (<= u 6.5e+157) (/ (- v) (+ u t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / (u * u)) * v;
	double tmp;
	if (u <= -1.3e+128) {
		tmp = t_1;
	} else if (u <= 6.5e+157) {
		tmp = -v / (u + t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t1 / (u * u)) * v
    if (u <= (-1.3d+128)) then
        tmp = t_1
    else if (u <= 6.5d+157) then
        tmp = -v / (u + t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 / (u * u)) * v;
	double tmp;
	if (u <= -1.3e+128) {
		tmp = t_1;
	} else if (u <= 6.5e+157) {
		tmp = -v / (u + t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / (u * u)) * v
	tmp = 0
	if u <= -1.3e+128:
		tmp = t_1
	elif u <= 6.5e+157:
		tmp = -v / (u + t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / Float64(u * u)) * v)
	tmp = 0.0
	if (u <= -1.3e+128)
		tmp = t_1;
	elseif (u <= 6.5e+157)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / (u * u)) * v;
	tmp = 0.0;
	if (u <= -1.3e+128)
		tmp = t_1;
	elseif (u <= 6.5e+157)
		tmp = -v / (u + t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision]}, If[LessEqual[u, -1.3e+128], t$95$1, If[LessEqual[u, 6.5e+157], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.3e128 or 6.5e157 < u

   1. Initial program 71.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 71.0

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

   5. Applied rewrites 71.0%

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

      1. lift-/.f64 N/A

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

      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 70.2

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

   7. Applied rewrites 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.2

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

   9. Applied rewrites 65.9%

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

   if -1.3e128 < u < 6.5e157

   1. Initial program 70.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 97.7

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

      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 97.7

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.8

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

   7. Applied rewrites 72.8%

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

Alternative 7: 62.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 66.0

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

7. Applied rewrites 66.0%

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

Alternative 8: 54.4% 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 70.8%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

Alternative 9: 14.3% accurate, 2.5× 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(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 70.8%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

7. Applied rewrites 59.1%

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

9. Applied rewrites 16.2%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))