Rosa's DopplerBench

Percentage Accurate: 73.3% → 98.0%

Time: 10.7s

Alternatives: 14

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{\frac{t1 + u}{0 - t1}}}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (/ (/ v (/ (+ t1 u) (- 0.0 t1))) (+ t1 u)))

C

double code(double u, double v, double t1) {
	return (v / ((t1 + u) / (0.0 - t1))) / (t1 + u);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / ((t1 + u) / (0.0d0 - t1))) / (t1 + u)
end function

Java

public static double code(double u, double v, double t1) {
	return (v / ((t1 + u) / (0.0 - t1))) / (t1 + u);
}

Python

def code(u, v, t1):
	return (v / ((t1 + u) / (0.0 - t1))) / (t1 + u)

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(Float64(t1 + u) / Float64(0.0 - t1))) / Float64(t1 + u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / ((t1 + u) / (0.0 - t1))) / (t1 + u);
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(N[(t1 + u), $MachinePrecision] / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

   11. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

   20. +-lowering-+.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

5. Simplified 98.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\frac{-1}{u + t1} \cdot t1\right)\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\frac{-1}{t1 + u} \cdot t1\right)\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   4. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{1}{\frac{t1 + u}{\mathsf{neg}\left(t1\right)}}\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{1}{\mathsf{neg}\left(\frac{t1 + u}{t1}\right)}\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v}{\mathsf{neg}\left(\frac{t1 + u}{t1}\right)}\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(v\right)\right)\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1}\right)}\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   9. frac-2neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(v\right)}{\frac{t1 + u}{t1}}\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(v\right)\right), \left(\frac{t1 + u}{t1}\right)\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - v\right), \left(\frac{t1 + u}{t1}\right)\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\frac{t1 + u}{t1}\right)\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   14. +-lowering-+.f64 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \frac{\color{blue}{\frac{0 - v}{\frac{t1 + u}{t1}}}}{u + t1} \]

8. Final simplification 98.8%

   \[\leadsto \frac{\frac{v}{\frac{t1 + u}{0 - t1}}}{t1 + u} \]
9. Add Preprocessing

Alternative 2: 77.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{u \cdot -2 - t1}{v}}\\ \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;\frac{0 - \frac{v \cdot t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ (- (* u -2.0) t1) v))))
   (if (<= t1 -1.65e-73)
     t_1
     (if (<= t1 2.9e+55) (/ (- 0.0 (/ (* v t1) u)) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 1.0 / (((u * -2.0) - t1) / v);
	double tmp;
	if (t1 <= -1.65e-73) {
		tmp = t_1;
	} else if (t1 <= 2.9e+55) {
		tmp = (0.0 - ((v * t1) / u)) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (((u * (-2.0d0)) - t1) / v)
    if (t1 <= (-1.65d-73)) then
        tmp = t_1
    else if (t1 <= 2.9d+55) then
        tmp = (0.0d0 - ((v * t1) / u)) / u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = 1.0 / (((u * -2.0) - t1) / v);
	double tmp;
	if (t1 <= -1.65e-73) {
		tmp = t_1;
	} else if (t1 <= 2.9e+55) {
		tmp = (0.0 - ((v * t1) / u)) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 1.0 / (((u * -2.0) - t1) / v)
	tmp = 0
	if t1 <= -1.65e-73:
		tmp = t_1
	elif t1 <= 2.9e+55:
		tmp = (0.0 - ((v * t1) / u)) / u
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(1.0 / Float64(Float64(Float64(u * -2.0) - t1) / v))
	tmp = 0.0
	if (t1 <= -1.65e-73)
		tmp = t_1;
	elseif (t1 <= 2.9e+55)
		tmp = Float64(Float64(0.0 - Float64(Float64(v * t1) / u)) / u);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 1.0 / (((u * -2.0) - t1) / v);
	tmp = 0.0;
	if (t1 <= -1.65e-73)
		tmp = t_1;
	elseif (t1 <= 2.9e+55)
		tmp = (0.0 - ((v * t1) / u)) / u;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(1.0 / N[(N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.65e-73], t$95$1, If[LessEqual[t1, 2.9e+55], N[(N[(0.0 - N[(N[(v * t1), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.65000000000000002e-73 or 2.8999999999999999e55 < t1

   1. Initial program 68.3%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot v\right), \color{blue}{\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)} \cdot \left(t1 + u\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(\color{blue}{\frac{t1 + u}{\mathsf{neg}\left(t1\right)}} \cdot \left(t1 + u\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)}\right), \color{blue}{\left(t1 + u\right)}\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      11. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \mathsf{+.f64}\left(t1, \color{blue}{u}\right)\right)\right) \]

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in u around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \color{blue}{\left(-2 \cdot u + -1 \cdot t1\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u + \left(\mathsf{neg}\left(t1\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u - \color{blue}{t1}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(-2 \cdot u\right), \color{blue}{t1}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right)\right) \]

      5. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

   7. Simplified 85.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{u \cdot -2 - t1}{1 \cdot v}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{u \cdot -2 - t1}{v}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(u \cdot -2 - t1\right), \color{blue}{v}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right), v\right)\right) \]

      6. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right), v\right)\right) \]

   9. Applied egg-rr 86.3%

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

   if -1.65000000000000002e-73 < t1 < 2.8999999999999999e55

   1. Initial program 90.3%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

      11. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

      20. +-lowering-+.f64 97.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

   5. Simplified 97.4%

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

   6. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{t1 \cdot v}{u}\right)}, \mathsf{+.f64}\left(u, t1\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{u}\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - t1 \cdot v\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      6. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t1, v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   8. Simplified 76.3%

      \[\leadsto \frac{\color{blue}{\frac{0 - t1 \cdot v}{u}}}{u + t1} \]

   9. Taylor expanded in u around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t1, v\right)\right), u\right), \color{blue}{u}\right) \]
   10. Step-by-step derivation

   11. Simplified 80.8%

       \[\leadsto \frac{\frac{0 - t1 \cdot v}{u}}{\color{blue}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{u \cdot -2 - t1}{v}}\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;\frac{0 - \frac{v \cdot t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u \cdot -2 - t1}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{u \cdot -2 - t1}{v}}\\ \mathbf{if}\;t1 \leq -1.95 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 0.042:\\ \;\;\;\;\frac{-1}{u \cdot \frac{u}{v \cdot t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ (- (* u -2.0) t1) v))))
   (if (<= t1 -1.95e-73)
     t_1
     (if (<= t1 0.042) (/ -1.0 (* u (/ u (* v t1)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 1.0 / (((u * -2.0) - t1) / v);
	double tmp;
	if (t1 <= -1.95e-73) {
		tmp = t_1;
	} else if (t1 <= 0.042) {
		tmp = -1.0 / (u * (u / (v * t1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (((u * (-2.0d0)) - t1) / v)
    if (t1 <= (-1.95d-73)) then
        tmp = t_1
    else if (t1 <= 0.042d0) then
        tmp = (-1.0d0) / (u * (u / (v * t1)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = 1.0 / (((u * -2.0) - t1) / v);
	double tmp;
	if (t1 <= -1.95e-73) {
		tmp = t_1;
	} else if (t1 <= 0.042) {
		tmp = -1.0 / (u * (u / (v * t1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 1.0 / (((u * -2.0) - t1) / v)
	tmp = 0
	if t1 <= -1.95e-73:
		tmp = t_1
	elif t1 <= 0.042:
		tmp = -1.0 / (u * (u / (v * t1)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(1.0 / Float64(Float64(Float64(u * -2.0) - t1) / v))
	tmp = 0.0
	if (t1 <= -1.95e-73)
		tmp = t_1;
	elseif (t1 <= 0.042)
		tmp = Float64(-1.0 / Float64(u * Float64(u / Float64(v * t1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 1.0 / (((u * -2.0) - t1) / v);
	tmp = 0.0;
	if (t1 <= -1.95e-73)
		tmp = t_1;
	elseif (t1 <= 0.042)
		tmp = -1.0 / (u * (u / (v * t1)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(1.0 / N[(N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.95e-73], t$95$1, If[LessEqual[t1, 0.042], N[(-1.0 / N[(u * N[(u / N[(v * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.94999999999999991e-73 or 0.0420000000000000026 < t1

   1. Initial program 68.9%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot v\right), \color{blue}{\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)} \cdot \left(t1 + u\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(\color{blue}{\frac{t1 + u}{\mathsf{neg}\left(t1\right)}} \cdot \left(t1 + u\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)}\right), \color{blue}{\left(t1 + u\right)}\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      11. +-lowering-+.f64 95.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \mathsf{+.f64}\left(t1, \color{blue}{u}\right)\right)\right) \]

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in u around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \color{blue}{\left(-2 \cdot u + -1 \cdot t1\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u + \left(\mathsf{neg}\left(t1\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u - \color{blue}{t1}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(-2 \cdot u\right), \color{blue}{t1}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right)\right) \]

      5. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

   7. Simplified 83.2%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{u \cdot -2 - t1}{1 \cdot v}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{u \cdot -2 - t1}{v}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(u \cdot -2 - t1\right), \color{blue}{v}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right), v\right)\right) \]

      6. *-lowering-*.f64 83.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right), v\right)\right) \]

   9. Applied egg-rr 83.9%

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

   if -1.94999999999999991e-73 < t1 < 0.0420000000000000026

   1. Initial program 92.6%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

      11. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

      20. +-lowering-+.f64 97.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{t1 \cdot v}{u}\right)}, \mathsf{+.f64}\left(u, t1\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{u}\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - t1 \cdot v\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      6. *-lowering-*.f64 78.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t1, v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   8. Simplified 78.4%

      \[\leadsto \frac{\color{blue}{\frac{0 - t1 \cdot v}{u}}}{u + t1} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}}} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}}\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}\right)\right)}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(u + t1\right) \cdot \frac{1}{\frac{0 - t1 \cdot v}{u}}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(u + t1\right) \cdot \frac{u}{0 - t1 \cdot v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{u}{0 - t1 \cdot v}\right)\right)}\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \left(\mathsf{neg}\left(\frac{u}{\mathsf{neg}\left(t1 \cdot v\right)}\right)\right)\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{u}{t1 \cdot v}\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \frac{u}{\color{blue}{t1 \cdot v}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(u + t1\right), \color{blue}{\left(\frac{u}{t1 \cdot v}\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(t1 + u\right), \left(\frac{\color{blue}{u}}{t1 \cdot v}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \left(\frac{\color{blue}{u}}{t1 \cdot v}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \color{blue}{\left(t1 \cdot v\right)}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \left(v \cdot \color{blue}{t1}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \mathsf{*.f64}\left(v, \color{blue}{t1}\right)\right)\right)\right) \]

   10. Applied egg-rr 77.7%

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

   11. Taylor expanded in t1 around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\color{blue}{u}, \mathsf{/.f64}\left(u, \mathsf{*.f64}\left(v, t1\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 82.8%

       \[\leadsto \frac{-1}{\color{blue}{u} \cdot \frac{u}{v \cdot t1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;\frac{-1}{u \cdot \frac{u}{v \cdot t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.9e-74)
     t_1
     (if (<= t1 2.9e+55) (/ -1.0 (* u (/ u (* v t1)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.9e-74) {
		tmp = t_1;
	} else if (t1 <= 2.9e+55) {
		tmp = -1.0 / (u * (u / (v * t1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.9d-74)) then
        tmp = t_1
    else if (t1 <= 2.9d+55) then
        tmp = (-1.0d0) / (u * (u / (v * t1)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.9e-74) {
		tmp = t_1;
	} else if (t1 <= 2.9e+55) {
		tmp = -1.0 / (u * (u / (v * t1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.9e-74:
		tmp = t_1
	elif t1 <= 2.9e+55:
		tmp = -1.0 / (u * (u / (v * t1)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.9e-74)
		tmp = t_1;
	elseif (t1 <= 2.9e+55)
		tmp = Float64(-1.0 / Float64(u * Float64(u / Float64(v * t1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.9e-74)
		tmp = t_1;
	elseif (t1 <= 2.9e+55)
		tmp = -1.0 / (u * (u / (v * t1)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.9e-74], t$95$1, If[LessEqual[t1, 2.9e+55], N[(-1.0 / N[(u * N[(u / N[(v * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.8999999999999998e-74 or 2.8999999999999999e55 < t1

   1. Initial program 68.3%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot v\right), \color{blue}{\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)} \cdot \left(t1 + u\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(\color{blue}{\frac{t1 + u}{\mathsf{neg}\left(t1\right)}} \cdot \left(t1 + u\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)}\right), \color{blue}{\left(t1 + u\right)}\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      11. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \mathsf{+.f64}\left(t1, \color{blue}{u}\right)\right)\right) \]

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in u around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \color{blue}{\left(-2 \cdot u + -1 \cdot t1\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u + \left(\mathsf{neg}\left(t1\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u - \color{blue}{t1}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(-2 \cdot u\right), \color{blue}{t1}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right)\right) \]

      5. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

   7. Simplified 85.9%

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

      1. *-lft-identity N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(u \cdot -2 - t1\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(u \cdot -2\right), \color{blue}{t1}\right)\right) \]

      4. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

   9. Applied egg-rr 85.9%

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

   if -1.8999999999999998e-74 < t1 < 2.8999999999999999e55

   1. Initial program 90.3%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

      11. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

      20. +-lowering-+.f64 97.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

   5. Simplified 97.4%

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

   6. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{t1 \cdot v}{u}\right)}, \mathsf{+.f64}\left(u, t1\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{u}\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - t1 \cdot v\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      6. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t1, v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   8. Simplified 76.3%

      \[\leadsto \frac{\color{blue}{\frac{0 - t1 \cdot v}{u}}}{u + t1} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}}} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}}\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}\right)\right)}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(u + t1\right) \cdot \frac{1}{\frac{0 - t1 \cdot v}{u}}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(u + t1\right) \cdot \frac{u}{0 - t1 \cdot v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{u}{0 - t1 \cdot v}\right)\right)}\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \left(\mathsf{neg}\left(\frac{u}{\mathsf{neg}\left(t1 \cdot v\right)}\right)\right)\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{u}{t1 \cdot v}\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \frac{u}{\color{blue}{t1 \cdot v}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(u + t1\right), \color{blue}{\left(\frac{u}{t1 \cdot v}\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(t1 + u\right), \left(\frac{\color{blue}{u}}{t1 \cdot v}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \left(\frac{\color{blue}{u}}{t1 \cdot v}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \color{blue}{\left(t1 \cdot v\right)}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \left(v \cdot \color{blue}{t1}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \mathsf{*.f64}\left(v, \color{blue}{t1}\right)\right)\right)\right) \]

   10. Applied egg-rr 75.6%

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

   11. Taylor expanded in t1 around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\color{blue}{u}, \mathsf{/.f64}\left(u, \mathsf{*.f64}\left(v, t1\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 80.1%

       \[\leadsto \frac{-1}{\color{blue}{u} \cdot \frac{u}{v \cdot t1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{v \cdot t1}{u \cdot \left(0 - u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.1e-73)
     t_1
     (if (<= t1 2.25e-5) (/ (* v t1) (* u (- 0.0 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.1e-73) {
		tmp = t_1;
	} else if (t1 <= 2.25e-5) {
		tmp = (v * t1) / (u * (0.0 - 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 * (-2.0d0)) - t1)
    if (t1 <= (-1.1d-73)) then
        tmp = t_1
    else if (t1 <= 2.25d-5) then
        tmp = (v * t1) / (u * (0.0d0 - 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 * -2.0) - t1);
	double tmp;
	if (t1 <= -1.1e-73) {
		tmp = t_1;
	} else if (t1 <= 2.25e-5) {
		tmp = (v * t1) / (u * (0.0 - u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.1e-73:
		tmp = t_1
	elif t1 <= 2.25e-5:
		tmp = (v * t1) / (u * (0.0 - u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.1e-73)
		tmp = t_1;
	elseif (t1 <= 2.25e-5)
		tmp = Float64(Float64(v * t1) / Float64(u * Float64(0.0 - u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.1e-73)
		tmp = t_1;
	elseif (t1 <= 2.25e-5)
		tmp = (v * t1) / (u * (0.0 - u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.1e-73], t$95$1, If[LessEqual[t1, 2.25e-5], N[(N[(v * t1), $MachinePrecision] / N[(u * N[(0.0 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.1e-73 or 2.25000000000000014e-5 < t1

   1. Initial program 68.9%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot v\right), \color{blue}{\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)} \cdot \left(t1 + u\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(\color{blue}{\frac{t1 + u}{\mathsf{neg}\left(t1\right)}} \cdot \left(t1 + u\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)}\right), \color{blue}{\left(t1 + u\right)}\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

      11. +-lowering-+.f64 95.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \mathsf{+.f64}\left(t1, \color{blue}{u}\right)\right)\right) \]

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in u around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \color{blue}{\left(-2 \cdot u + -1 \cdot t1\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u + \left(\mathsf{neg}\left(t1\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u - \color{blue}{t1}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(-2 \cdot u\right), \color{blue}{t1}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right)\right) \]

      5. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

   7. Simplified 83.2%

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

      1. *-lft-identity N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(u \cdot -2 - t1\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(u \cdot -2\right), \color{blue}{t1}\right)\right) \]

      4. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

   9. Applied egg-rr 83.2%

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

   if -1.1e-73 < t1 < 2.25000000000000014e-5

   1. Initial program 92.6%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \color{blue}{\left({u}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \left(u \cdot \color{blue}{u}\right)\right) \]

      2. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(u, \color{blue}{u}\right)\right) \]

   5. Simplified 80.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{v \cdot t1}{u \cdot \left(0 - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\frac{u}{v}}\\ \mathbf{if}\;u \leq -3 \cdot 10^{+237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ -1.0 (/ u v))))
   (if (<= u -3e+237) t_1 (if (<= u 5.8e+153) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 / (u / v);
	double tmp;
	if (u <= -3e+237) {
		tmp = t_1;
	} else if (u <= 5.8e+153) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-1.0d0) / (u / v)
    if (u <= (-3d+237)) then
        tmp = t_1
    else if (u <= 5.8d+153) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -1.0 / (u / v);
	double tmp;
	if (u <= -3e+237) {
		tmp = t_1;
	} else if (u <= 5.8e+153) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 / (u / v)
	tmp = 0
	if u <= -3e+237:
		tmp = t_1
	elif u <= 5.8e+153:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 / Float64(u / v))
	tmp = 0.0
	if (u <= -3e+237)
		tmp = t_1;
	elseif (u <= 5.8e+153)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 / (u / v);
	tmp = 0.0;
	if (u <= -3e+237)
		tmp = t_1;
	elseif (u <= 5.8e+153)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3e+237], t$95$1, If[LessEqual[u, 5.8e+153], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3e237 or 5.80000000000000004e153 < u

   1. Initial program 83.0%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

      11. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

      20. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{t1 \cdot v}{u}\right)}, \mathsf{+.f64}\left(u, t1\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{u}\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - t1 \cdot v\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(t1 \cdot v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

      6. *-lowering-*.f64 94.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t1, v\right)\right), u\right), \mathsf{+.f64}\left(u, t1\right)\right) \]

   8. Simplified 94.1%

      \[\leadsto \frac{\color{blue}{\frac{0 - t1 \cdot v}{u}}}{u + t1} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}}} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}}\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\frac{u + t1}{\frac{0 - t1 \cdot v}{u}}\right)\right)}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(u + t1\right) \cdot \frac{1}{\frac{0 - t1 \cdot v}{u}}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(u + t1\right) \cdot \frac{u}{0 - t1 \cdot v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{u}{0 - t1 \cdot v}\right)\right)}\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \left(\mathsf{neg}\left(\frac{u}{\mathsf{neg}\left(t1 \cdot v\right)}\right)\right)\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{u}{t1 \cdot v}\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(u + t1\right) \cdot \frac{u}{\color{blue}{t1 \cdot v}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(u + t1\right), \color{blue}{\left(\frac{u}{t1 \cdot v}\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(t1 + u\right), \left(\frac{\color{blue}{u}}{t1 \cdot v}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \left(\frac{\color{blue}{u}}{t1 \cdot v}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \color{blue}{\left(t1 \cdot v\right)}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \left(v \cdot \color{blue}{t1}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \mathsf{/.f64}\left(u, \mathsf{*.f64}\left(v, \color{blue}{t1}\right)\right)\right)\right) \]

   10. Applied egg-rr 94.0%

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

   11. Taylor expanded in t1 around inf

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{u}{v}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 55.5%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(u, \color{blue}{v}\right)\right) \]

   13. Simplified 55.5%

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

   if -3e237 < u < 5.80000000000000004e153

   1. Initial program 77.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot t1\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(t1\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \left(0 - \color{blue}{t1}\right)\right) \]

      7. --lowering--.f64 62.7%

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \color{blue}{t1}\right)\right) \]

   5. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(t1\right)\right)\right) \]

      2. neg-lowering-neg.f64 62.7%

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{neg.f64}\left(t1\right)\right) \]

   7. Applied egg-rr 62.7%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+237}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u} \cdot -0.5\\ \mathbf{if}\;u \leq -3 \cdot 10^{+237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v u) -0.5)))
   (if (<= u -3e+237) t_1 (if (<= u 2.9e+154) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) * -0.5;
	double tmp;
	if (u <= -3e+237) {
		tmp = t_1;
	} else if (u <= 2.9e+154) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / u) * (-0.5d0)
    if (u <= (-3d+237)) then
        tmp = t_1
    else if (u <= 2.9d+154) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / u) * -0.5;
	double tmp;
	if (u <= -3e+237) {
		tmp = t_1;
	} else if (u <= 2.9e+154) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) * -0.5
	tmp = 0
	if u <= -3e+237:
		tmp = t_1
	elif u <= 2.9e+154:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) * -0.5)
	tmp = 0.0
	if (u <= -3e+237)
		tmp = t_1;
	elseif (u <= 2.9e+154)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / u) * -0.5;
	tmp = 0.0;
	if (u <= -3e+237)
		tmp = t_1;
	elseif (u <= 2.9e+154)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[u, -3e+237], t$95$1, If[LessEqual[u, 2.9e+154], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3e237 or 2.89999999999999979e154 < u

   1. Initial program 83.0%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \color{blue}{\left(2 \cdot \left(t1 \cdot u\right) + {u}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \left({u}^{2} + \color{blue}{2 \cdot \left(t1 \cdot u\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \left(u \cdot u + \color{blue}{2} \cdot \left(t1 \cdot u\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \left(u \cdot u + \left(2 \cdot t1\right) \cdot \color{blue}{u}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \left(u \cdot \color{blue}{\left(u + 2 \cdot t1\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(u, \color{blue}{\left(u + 2 \cdot t1\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(u, \mathsf{+.f64}\left(u, \color{blue}{\left(2 \cdot t1\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 83.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(u, \mathsf{+.f64}\left(u, \mathsf{*.f64}\left(2, \color{blue}{t1}\right)\right)\right)\right) \]

   5. Simplified 83.0%

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

   6. Taylor expanded in t1 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{v}{u}\right), \color{blue}{\frac{-1}{2}}\right) \]

      3. /-lowering-/.f64 52.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(v, u\right), \frac{-1}{2}\right) \]

   8. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   if -3e237 < u < 2.89999999999999979e154

   1. Initial program 77.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot t1\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(t1\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \left(0 - \color{blue}{t1}\right)\right) \]

      7. --lowering--.f64 62.7%

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \color{blue}{t1}\right)\right) \]

   5. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(t1\right)\right)\right) \]

      2. neg-lowering-neg.f64 62.7%

         \[\leadsto \mathsf{/.f64}\left(v, \mathsf{neg.f64}\left(t1\right)\right) \]

   7. Applied egg-rr 62.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+237}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v * (t1 * (-1.0 / (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 = (v * (t1 * ((-1.0d0) / (t1 + u)))) / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

   11. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

   20. +-lowering-+.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 9: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{0 - \left(t1 + u\right)}}{\frac{t1 + u}{t1}} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (/ (/ v (- 0.0 (+ t1 u))) (/ (+ t1 u) t1)))

C

double code(double u, double v, double t1) {
	return (v / (0.0 - (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 / (0.0d0 - (t1 + u))) / ((t1 + u) / t1)
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (0.0 - (t1 + u))) / ((t1 + u) / t1);
}

Python

def code(u, v, t1):
	return (v / (0.0 - (t1 + u))) / ((t1 + u) / t1)

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(0.0 - Float64(t1 + u))) / Float64(Float64(t1 + u) / t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (0.0 - (t1 + u))) / ((t1 + u) / t1);
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(0.0 - N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v}{t1 + u}\right), \color{blue}{\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(t1 + u\right)\right), \left(\frac{\color{blue}{t1 + u}}{\mathsf{neg}\left(t1\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \left(\frac{t1 + \color{blue}{u}}{\mathsf{neg}\left(t1\right)}\right)\right) \]

   8. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \left(\mathsf{neg}\left(\frac{t1 + u}{t1}\right)\right)\right) \]

   9. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{neg.f64}\left(\left(\frac{t1 + u}{t1}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right)\right) \]

   11. +-lowering-+.f64 97.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right)\right) \]

4. Applied egg-rr 97.0%

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

5. Final simplification 97.0%

   \[\leadsto \frac{\frac{v}{0 - \left(t1 + u\right)}}{\frac{t1 + u}{t1}} \]
6. Add Preprocessing

Alternative 10: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1 + u} \cdot \frac{t1}{0 - \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (* (/ v (+ t1 u)) (/ t1 (- 0.0 (+ t1 u)))))

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) * (t1 / (0.0 - (t1 + u)));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (t1 + u)) * (t1 / (0.0d0 - (t1 + u)))
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (t1 + u)) * (t1 / (0.0 - (t1 + u)));
}

Python

def code(u, v, t1):
	return (v / (t1 + u)) * (t1 / (0.0 - (t1 + u)))

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(0.0 - Float64(t1 + u))))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (t1 + u)) * (t1 / (0.0 - (t1 + u)));
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / N[(0.0 - N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

   11. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

   20. +-lowering-+.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

5. Simplified 98.7%

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. distribute-frac-neg N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{t1 + u} \cdot \frac{t1}{t1 + u}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{v}{t1 + u}\right), \left(\frac{t1}{t1 + u}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \left(t1 + u\right)\right), \left(\frac{t1}{t1 + u}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \left(\frac{t1}{t1 + u}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{/.f64}\left(t1, \left(t1 + u\right)\right)\right)\right) \]

   16. +-lowering-+.f64 97.0%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{/.f64}\left(t1, \mathsf{+.f64}\left(t1, u\right)\right)\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Final simplification 97.0%

   \[\leadsto \frac{v}{t1 + u} \cdot \frac{t1}{0 - \left(t1 + u\right)} \]
9. Add Preprocessing

Alternative 11: 61.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

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

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - 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 * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

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

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot v\right), \color{blue}{\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)} \cdot \left(t1 + u\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(\color{blue}{\frac{t1 + u}{\mathsf{neg}\left(t1\right)}} \cdot \left(t1 + u\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\frac{t1 + u}{\mathsf{neg}\left(t1\right)}\right), \color{blue}{\left(t1 + u\right)}\right)\right) \]

   7. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

   8. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\frac{t1 + u}{t1}\right)\right), \left(\color{blue}{t1} + u\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(t1 + u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \left(t1 + u\right)\right)\right) \]

   11. +-lowering-+.f64 94.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), t1\right)\right), \mathsf{+.f64}\left(t1, \color{blue}{u}\right)\right)\right) \]

4. Applied egg-rr 94.6%

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

5. Taylor expanded in u around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \color{blue}{\left(-2 \cdot u + -1 \cdot t1\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u + \left(\mathsf{neg}\left(t1\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \left(-2 \cdot u - \color{blue}{t1}\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(-2 \cdot u\right), \color{blue}{t1}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\left(u \cdot -2\right), t1\right)\right) \]

   5. *-lowering-*.f64 63.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, v\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

7. Simplified 63.8%

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

   1. *-lft-identity N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(u \cdot -2 - t1\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(u \cdot -2\right), \color{blue}{t1}\right)\right) \]

   4. *-lowering-*.f64 63.8%

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(u, -2\right), t1\right)\right) \]

9. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{\frac{v}{u \cdot -2 - t1}} \]
10. Add Preprocessing

Alternative 12: 61.1% accurate, 1.7× speedup

Math

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

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(-1.0 / Float64(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[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

   11. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

   20. +-lowering-+.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

5. Simplified 98.7%

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

6. Taylor expanded in t1 around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot v\right)}, \mathsf{+.f64}\left(u, t1\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(v\right)\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - v\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   3. --lowering--.f64 62.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

8. Simplified 62.9%

   \[\leadsto \frac{\color{blue}{0 - v}}{u + t1} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{u + t1}{0 - v}}} \]

   2. frac-2neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{u + t1}{0 - v}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{u + t1}{0 - v}}\right)} \]

   4. sub0-neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. remove-double-neg N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{u + t1}{v}\right)}\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\left(u + t1\right), \color{blue}{v}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\left(t1 + u\right), v\right)\right) \]

   10. +-lowering-+.f64 63.4%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(t1, u\right), v\right)\right) \]

10. Applied egg-rr 63.4%

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

Alternative 13: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{0 - \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- 0.0 (+ t1 u))))

C

double code(double u, double v, double t1) {
	return v / (0.0 - (t1 + u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (0.0d0 - (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return v / (0.0 - (t1 + u));
}

Python

def code(u, v, t1):
	return v / (0.0 - (t1 + u))

Julia

function code(u, v, t1)
	return Float64(v / Float64(0.0 - Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (0.0 - (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(0.0 - N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(t1 \cdot v\right)}{t1 + u}\right), \color{blue}{\left(t1 + u\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot t1\right) \cdot v}{t1 + u}\right), \left(t1 + u\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot \left(-1 \cdot t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{-1 \cdot t1}{t1 + u}\right), \left(\color{blue}{t1} + u\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right), \left(t1 + u\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(\frac{t1}{t1 + u}\right)\right)\right), \left(\color{blue}{t1} + u\right)\right) \]

   11. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\mathsf{neg}\left(t1\right)}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{-1 \cdot t1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1 \cdot -1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(t1 \cdot \frac{-1}{t1 + u}\right)\right), \left(t1 + u\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \left(\frac{-1}{t1 + u}\right)\right)\right), \left(t1 + u\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(t1 + u\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \left(u + t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(t1 + u\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \left(u + \color{blue}{t1}\right)\right) \]

   20. +-lowering-+.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(t1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(u, t1\right)\right)\right)\right), \mathsf{+.f64}\left(u, \color{blue}{t1}\right)\right) \]

5. Simplified 98.7%

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

6. Taylor expanded in t1 around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot v\right)}, \mathsf{+.f64}\left(u, t1\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(v\right)\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - v\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   3. --lowering--.f64 62.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

8. Simplified 62.9%

   \[\leadsto \frac{\color{blue}{0 - v}}{u + t1} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(v\right)\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

   2. neg-lowering-neg.f64 62.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(v\right), \mathsf{+.f64}\left(\color{blue}{u}, t1\right)\right) \]

10. Applied egg-rr 62.9%

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

11. Final simplification 62.9%

    \[\leadsto \frac{v}{0 - \left(t1 + u\right)} \]
12. Add Preprocessing

Alternative 14: 53.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{v}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (- 0.0 (/ v t1)))

C

double code(double u, double v, double t1) {
	return 0.0 - (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 = 0.0d0 - (v / t1)
end function

Java

public static double code(double u, double v, double t1) {
	return 0.0 - (v / t1);
}

Python

def code(u, v, t1):
	return 0.0 - (v / t1)

Julia

function code(u, v, t1)
	return Float64(0.0 - Float64(v / t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = 0.0 - (v / t1);
end

Wolfram

code[u_, v_, t1_] := N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot t1\right)}\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(t1\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \left(0 - \color{blue}{t1}\right)\right) \]

   7. --lowering--.f64 55.1%

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \color{blue}{t1}\right)\right) \]

5. Simplified 55.1%

   \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
6. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(t1\right)\right)\right) \]

   2. neg-lowering-neg.f64 55.1%

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{neg.f64}\left(t1\right)\right) \]

7. Applied egg-rr 55.1%

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

8. Final simplification 55.1%

   \[\leadsto 0 - \frac{v}{t1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))