Rosa's DopplerBench

Percentage Accurate: 72.1% → 98.3%

Time: 10.4s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. *-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. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

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

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

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

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

   9. neg-sub0 N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 98.3%

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

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

Alternative 2: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2.45 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-129}:\\ \;\;\;\;\frac{v}{0 - \frac{u}{\frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (+ t1 u)))))
   (if (<= t1 -2.45e-49)
     t_1
     (if (<= t1 3e-129) (/ v (- 0.0 (/ u (/ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -2.45e-49) {
		tmp = t_1;
	} else if (t1 <= 3e-129) {
		tmp = v / (0.0 - (u / (t1 / u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0d0 - (v / (t1 + u))
    if (t1 <= (-2.45d-49)) then
        tmp = t_1
    else if (t1 <= 3d-129) then
        tmp = v / (0.0d0 - (u / (t1 / u)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -2.45e-49) {
		tmp = t_1;
	} else if (t1 <= 3e-129) {
		tmp = v / (0.0 - (u / (t1 / u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (v / (t1 + u))
	tmp = 0
	if t1 <= -2.45e-49:
		tmp = t_1
	elif t1 <= 3e-129:
		tmp = v / (0.0 - (u / (t1 / u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -2.45e-49)
		tmp = t_1;
	elseif (t1 <= 3e-129)
		tmp = Float64(v / Float64(0.0 - Float64(u / Float64(t1 / u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 0.0 - (v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -2.45e-49)
		tmp = t_1;
	elseif (t1 <= 3e-129)
		tmp = v / (0.0 - (u / (t1 / u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.45e-49], t$95$1, If[LessEqual[t1, 3e-129], N[(v / N[(0.0 - N[(u / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.4500000000000001e-49 or 2.9999999999999998e-129 < t1

   1. Initial program 67.8%

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

      1. *-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. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

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

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sub0-neg N/A

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

      4. distribute-frac-neg N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf

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

   9. Simplified 83.9%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. /-rgt-identity N/A

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

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

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

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

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

       6. +-lowering-+.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

   if -2.4500000000000001e-49 < t1 < 2.9999999999999998e-129

   1. Initial program 75.2%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{t1 \cdot v}{{u}^{2}}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{t1 \cdot v}{{u}^{2}}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{v \cdot t1}{{\color{blue}{u}}^{2}}\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(v \cdot \color{blue}{\frac{t1}{{u}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \color{blue}{\left(\frac{t1}{{u}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \color{blue}{\left({u}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.8%

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

   5. Simplified 77.8%

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

      1. cancel-sign-sub-inv N/A

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

      2. +-lft-identity N/A

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

      3. sub0-neg N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. frac-2neg N/A

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

      7. sub0-neg N/A

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

      8. remove-double-neg N/A

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

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

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

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

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

      11. clear-num N/A

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

      12. associate-/r* N/A

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

      13. clear-num N/A

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

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

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

      15. /-lowering-/.f64 91.4%

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

   7. Applied egg-rr 91.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.45 \cdot 10^{-49}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-129}:\\ \;\;\;\;\frac{v}{0 - \frac{u}{\frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{0 - t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (+ t1 u)))))
   (if (<= t1 -1.8e-47)
     t_1
     (if (<= t1 3.9e-128) (/ v (* u (/ u (- 0.0 t1)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -1.8e-47) {
		tmp = t_1;
	} else if (t1 <= 3.9e-128) {
		tmp = v / (u * (u / (0.0 - 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 = 0.0d0 - (v / (t1 + u))
    if (t1 <= (-1.8d-47)) then
        tmp = t_1
    else if (t1 <= 3.9d-128) then
        tmp = v / (u * (u / (0.0d0 - 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 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -1.8e-47) {
		tmp = t_1;
	} else if (t1 <= 3.9e-128) {
		tmp = v / (u * (u / (0.0 - t1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (v / (t1 + u))
	tmp = 0
	if t1 <= -1.8e-47:
		tmp = t_1
	elif t1 <= 3.9e-128:
		tmp = v / (u * (u / (0.0 - t1)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -1.8e-47)
		tmp = t_1;
	elseif (t1 <= 3.9e-128)
		tmp = Float64(v / Float64(u * Float64(u / Float64(0.0 - t1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 0.0 - (v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -1.8e-47)
		tmp = t_1;
	elseif (t1 <= 3.9e-128)
		tmp = v / (u * (u / (0.0 - t1)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e-47], t$95$1, If[LessEqual[t1, 3.9e-128], N[(v / N[(u * N[(u / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.79999999999999995e-47 or 3.89999999999999997e-128 < t1

   1. Initial program 67.8%

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

      1. *-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. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

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

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sub0-neg N/A

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

      4. distribute-frac-neg N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf

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

   9. Simplified 83.9%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. /-rgt-identity N/A

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

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

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

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

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

       6. +-lowering-+.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

   if -1.79999999999999995e-47 < t1 < 3.89999999999999997e-128

   1. Initial program 75.2%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{t1 \cdot v}{{u}^{2}}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{t1 \cdot v}{{u}^{2}}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{v \cdot t1}{{\color{blue}{u}}^{2}}\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(v \cdot \color{blue}{\frac{t1}{{u}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \color{blue}{\left(\frac{t1}{{u}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \color{blue}{\left({u}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.8%

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

   5. Simplified 77.8%

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

      1. cancel-sign-sub-inv N/A

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

      2. +-lft-identity N/A

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

      3. sub0-neg N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. frac-2neg N/A

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

      7. sub0-neg N/A

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

      8. remove-double-neg N/A

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

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

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

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

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

      11. clear-num N/A

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

      12. associate-/r* N/A

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

      13. clear-num N/A

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

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

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

      15. /-lowering-/.f64 91.4%

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

   7. Applied egg-rr 91.4%

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

      1. associate-/r/ N/A

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

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

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

      3. /-lowering-/.f64 91.2%

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

   9. Applied egg-rr 91.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{-47}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{0 - t1}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{0 - u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (+ t1 u)))))
   (if (<= t1 -7.5e-50)
     t_1
     (if (<= t1 4.5e-129) (* (/ t1 u) (/ v (- 0.0 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -7.5e-50) {
		tmp = t_1;
	} else if (t1 <= 4.5e-129) {
		tmp = (t1 / u) * (v / (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 = 0.0d0 - (v / (t1 + u))
    if (t1 <= (-7.5d-50)) then
        tmp = t_1
    else if (t1 <= 4.5d-129) then
        tmp = (t1 / u) * (v / (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 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -7.5e-50) {
		tmp = t_1;
	} else if (t1 <= 4.5e-129) {
		tmp = (t1 / u) * (v / (0.0 - u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (v / (t1 + u))
	tmp = 0
	if t1 <= -7.5e-50:
		tmp = t_1
	elif t1 <= 4.5e-129:
		tmp = (t1 / u) * (v / (0.0 - u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -7.5e-50)
		tmp = t_1;
	elseif (t1 <= 4.5e-129)
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(0.0 - u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 0.0 - (v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -7.5e-50)
		tmp = t_1;
	elseif (t1 <= 4.5e-129)
		tmp = (t1 / u) * (v / (0.0 - u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.5e-50], t$95$1, If[LessEqual[t1, 4.5e-129], N[(N[(t1 / u), $MachinePrecision] * N[(v / N[(0.0 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.5e-50 or 4.50000000000000031e-129 < t1

   1. Initial program 67.8%

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

      1. *-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. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

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

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sub0-neg N/A

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

      4. distribute-frac-neg N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf

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

   9. Simplified 83.9%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. /-rgt-identity N/A

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

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

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

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

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

       6. +-lowering-+.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

   if -7.5e-50 < t1 < 4.50000000000000031e-129

   1. Initial program 75.2%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{t1 \cdot v}{{u}^{2}}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{t1 \cdot v}{{u}^{2}}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{v \cdot t1}{{\color{blue}{u}}^{2}}\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(v \cdot \color{blue}{\frac{t1}{{u}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \color{blue}{\left(\frac{t1}{{u}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \color{blue}{\left({u}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.8%

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

   5. Simplified 77.8%

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

      1. sub0-neg N/A

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

      2. associate-*r/ N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-50}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{0 - u}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{v}{u}}{-1}\\ \mathbf{if}\;u \leq -3.1 \cdot 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+127}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (/ v u) -1.0)))
   (if (<= u -3.1e+209) t_1 (if (<= u 1.8e+127) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) / -1.0;
	double tmp;
	if (u <= -3.1e+209) {
		tmp = t_1;
	} else if (u <= 1.8e+127) {
		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) / (-1.0d0)
    if (u <= (-3.1d+209)) then
        tmp = t_1
    else if (u <= 1.8d+127) 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) / -1.0;
	double tmp;
	if (u <= -3.1e+209) {
		tmp = t_1;
	} else if (u <= 1.8e+127) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) / -1.0
	tmp = 0
	if u <= -3.1e+209:
		tmp = t_1
	elif u <= 1.8e+127:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) / -1.0)
	tmp = 0.0
	if (u <= -3.1e+209)
		tmp = t_1;
	elseif (u <= 1.8e+127)
		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) / -1.0;
	tmp = 0.0;
	if (u <= -3.1e+209)
		tmp = t_1;
	elseif (u <= 1.8e+127)
		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] / -1.0), $MachinePrecision]}, If[LessEqual[u, -3.1e+209], t$95$1, If[LessEqual[u, 1.8e+127], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.1000000000000001e209 or 1.79999999999999989e127 < u

   1. Initial program 79.0%

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

      1. *-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. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

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

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sub0-neg N/A

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

      4. distribute-frac-neg N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in t1 around inf

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

   9. Simplified 51.4%

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

   10. Taylor expanded in t1 around 0

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

       1. /-lowering-/.f64 48.7%

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

   12. Simplified 48.7%

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

   if -3.1000000000000001e209 < u < 1.79999999999999989e127

   1. Initial program 68.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 66.0%

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

   5. Simplified 66.0%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 66.0%

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

   7. Applied egg-rr 66.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+127}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. *-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. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

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

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

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

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

   9. neg-sub0 N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. sub0-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-frac-neg N/A

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

   8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

   14. +-lowering-+.f64 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in t1 around 0

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

   1. div-sub N/A

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

   2. associate-/l* N/A

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

   3. *-inverses N/A

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

   4. metadata-eval N/A

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

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

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

   6. /-lowering-/.f64 98.3%

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

9. Simplified 98.3%

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

Alternative 7: 62.1% 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 70.5%

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

   1. *-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. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

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

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

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

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

   9. neg-sub0 N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. sub0-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-frac-neg N/A

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

   8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

   14. +-lowering-+.f64 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in t1 around inf

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

9. Simplified 64.5%

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

    1. div-inv N/A

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

    2. metadata-eval N/A

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

    3. *-commutative N/A

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

    4. clear-num N/A

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

    5. div-inv N/A

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

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

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

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

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

    8. +-lowering-+.f64 64.7%

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

11. Applied egg-rr 64.7%

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

Alternative 8: 62.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. *-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. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

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

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

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

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

   9. neg-sub0 N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. sub0-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-frac-neg N/A

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

   8. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

   14. +-lowering-+.f64 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in t1 around inf

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

9. Simplified 64.5%

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

    1. frac-2neg N/A

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

    2. metadata-eval N/A

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

    3. /-rgt-identity N/A

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

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

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

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

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

    6. +-lowering-+.f64 64.5%

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

11. Applied egg-rr 64.5%

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

12. Final simplification 64.5%

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

Alternative 9: 54.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return -1.0 / (t1 / 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 / v)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. neg-sub0 N/A

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

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

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

   4. /-lowering-/.f64 57.2%

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

5. Simplified 57.2%

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

   1. sub0-neg N/A

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

   2. clear-num N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

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

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

   6. /-lowering-/.f64 57.3%

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

7. Applied egg-rr 57.3%

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

Alternative 10: 55.1% 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 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. neg-sub0 N/A

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

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

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

   4. /-lowering-/.f64 57.2%

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

5. Simplified 57.2%

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

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 57.2%

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

7. Applied egg-rr 57.2%

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

8. Final simplification 57.2%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))