Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.7%

Time: 11.3s

Alternatives: 12

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: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. 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

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 97.8%

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

4. Applied egg-rr 97.8%

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

Alternative 2: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-100}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.6e+39)
   (/ (/ v (+ t1 u)) -1.0)
   (if (<= t1 1.75e-100)
     (/ (- 0.0 (/ t1 u)) (/ u v))
     (/ (/ v t1) (- -1.0 (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.6e+39) {
		tmp = (v / (t1 + u)) / -1.0;
	} else if (t1 <= 1.75e-100) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = (v / t1) / (-1.0 - (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.6d+39)) then
        tmp = (v / (t1 + u)) / (-1.0d0)
    else if (t1 <= 1.75d-100) then
        tmp = (0.0d0 - (t1 / u)) / (u / v)
    else
        tmp = (v / t1) / ((-1.0d0) - (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.6e+39) {
		tmp = (v / (t1 + u)) / -1.0;
	} else if (t1 <= 1.75e-100) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = (v / t1) / (-1.0 - (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.6e+39:
		tmp = (v / (t1 + u)) / -1.0
	elif t1 <= 1.75e-100:
		tmp = (0.0 - (t1 / u)) / (u / v)
	else:
		tmp = (v / t1) / (-1.0 - (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.6e+39)
		tmp = Float64(Float64(v / Float64(t1 + u)) / -1.0);
	elseif (t1 <= 1.75e-100)
		tmp = Float64(Float64(0.0 - Float64(t1 / u)) / Float64(u / v));
	else
		tmp = Float64(Float64(v / t1) / Float64(-1.0 - Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.6e+39)
		tmp = (v / (t1 + u)) / -1.0;
	elseif (t1 <= 1.75e-100)
		tmp = (0.0 - (t1 / u)) / (u / v);
	else
		tmp = (v / t1) / (-1.0 - (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.6e+39], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision], If[LessEqual[t1, 1.75e-100], N[(N[(0.0 - N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(N[(v / t1), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.59999999999999996e39

   1. Initial program 61.4%

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

      1. *-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 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 88.6%

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

   if -1.59999999999999996e39 < t1 < 1.75e-100

   1. Initial program 77.9%

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

      1. *-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 95.1%

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 74.0%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 79.1%

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

       1. div-inv N/A

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

       2. clear-num N/A

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

       3. associate-*l/ N/A

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

       4. div-inv N/A

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

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

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

       6. distribute-neg-frac N/A

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

       7. clear-num N/A

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

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

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

       9. neg-sub0 N/A

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

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

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

       11. /-lowering-/.f64 80.7%

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

   12. Applied egg-rr 80.7%

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

   if 1.75e-100 < t1

   1. Initial program 73.9%

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

      1. *-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 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t1 around inf

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

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

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

      2. /-lowering-/.f64 99.8%

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

   7. Simplified 99.8%

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

      1. /-lowering-/.f64 N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

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

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

      8. /-lowering-/.f64 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in t1 around inf

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

       1. /-lowering-/.f64 78.6%

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

   12. Simplified 78.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-100}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.76 \cdot 10^{-99}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (/ v (+ t1 u)) -1.0)))
   (if (<= t1 -1.6e+39)
     t_1
     (if (<= t1 1.76e-99) (/ (- 0.0 (/ t1 u)) (/ u v)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) / -1.0;
	double tmp;
	if (t1 <= -1.6e+39) {
		tmp = t_1;
	} else if (t1 <= 1.76e-99) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / (t1 + u)) / (-1.0d0)
    if (t1 <= (-1.6d+39)) then
        tmp = t_1
    else if (t1 <= 1.76d-99) then
        tmp = (0.0d0 - (t1 / u)) / (u / v)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) / -1.0;
	double tmp;
	if (t1 <= -1.6e+39) {
		tmp = t_1;
	} else if (t1 <= 1.76e-99) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (t1 + u)) / -1.0
	tmp = 0
	if t1 <= -1.6e+39:
		tmp = t_1
	elif t1 <= 1.76e-99:
		tmp = (0.0 - (t1 / u)) / (u / v)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) / -1.0)
	tmp = 0.0
	if (t1 <= -1.6e+39)
		tmp = t_1;
	elseif (t1 <= 1.76e-99)
		tmp = Float64(Float64(0.0 - Float64(t1 / u)) / Float64(u / v));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / (t1 + u)) / -1.0;
	tmp = 0.0;
	if (t1 <= -1.6e+39)
		tmp = t_1;
	elseif (t1 <= 1.76e-99)
		tmp = (0.0 - (t1 / u)) / (u / v);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]}, If[LessEqual[t1, -1.6e+39], t$95$1, If[LessEqual[t1, 1.76e-99], N[(N[(0.0 - N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999996e39 or 1.75999999999999991e-99 < t1

   1. Initial program 69.4%

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

      1. *-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 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 81.5%

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

   if -1.59999999999999996e39 < t1 < 1.75999999999999991e-99

   1. Initial program 77.9%

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

      1. *-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 95.1%

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 74.0%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 79.1%

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

       1. div-inv N/A

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

       2. clear-num N/A

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

       3. associate-*l/ N/A

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

       4. div-inv N/A

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

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

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

       6. distribute-neg-frac N/A

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

       7. clear-num N/A

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

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

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

       9. neg-sub0 N/A

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

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

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

       11. /-lowering-/.f64 80.7%

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

   12. Applied egg-rr 80.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{elif}\;t1 \leq 1.76 \cdot 10^{-99}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{0 - t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (/ v (+ t1 u)) -1.0)))
   (if (<= t1 -1.6e+39)
     t_1
     (if (<= t1 3.9e-99) (/ (/ v u) (/ u (- 0.0 t1))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) / -1.0;
	double tmp;
	if (t1 <= -1.6e+39) {
		tmp = t_1;
	} else if (t1 <= 3.9e-99) {
		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 = (v / (t1 + u)) / (-1.0d0)
    if (t1 <= (-1.6d+39)) then
        tmp = t_1
    else if (t1 <= 3.9d-99) 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 = (v / (t1 + u)) / -1.0;
	double tmp;
	if (t1 <= -1.6e+39) {
		tmp = t_1;
	} else if (t1 <= 3.9e-99) {
		tmp = (v / u) / (u / (0.0 - t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (t1 + u)) / -1.0
	tmp = 0
	if t1 <= -1.6e+39:
		tmp = t_1
	elif t1 <= 3.9e-99:
		tmp = (v / u) / (u / (0.0 - t1))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) / -1.0)
	tmp = 0.0
	if (t1 <= -1.6e+39)
		tmp = t_1;
	elseif (t1 <= 3.9e-99)
		tmp = Float64(Float64(v / 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 = (v / (t1 + u)) / -1.0;
	tmp = 0.0;
	if (t1 <= -1.6e+39)
		tmp = t_1;
	elseif (t1 <= 3.9e-99)
		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[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]}, If[LessEqual[t1, -1.6e+39], t$95$1, If[LessEqual[t1, 3.9e-99], N[(N[(v / u), $MachinePrecision] / N[(u / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999996e39 or 3.89999999999999987e-99 < t1

   1. Initial program 69.4%

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

      1. *-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 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 81.5%

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

   if -1.59999999999999996e39 < t1 < 3.89999999999999987e-99

   1. Initial program 77.9%

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

      1. *-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 95.1%

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 74.0%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 79.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{0 - t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{0 - u \cdot u}\\ \mathbf{if}\;u \leq -5.2 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+66}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v (- 0.0 (* u u))))))
   (if (<= u -5.2e+21) t_1 (if (<= u 5e+66) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (0.0 - (u * u)));
	double tmp;
	if (u <= -5.2e+21) {
		tmp = t_1;
	} else if (u <= 5e+66) {
		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 = t1 * (v / (0.0d0 - (u * u)))
    if (u <= (-5.2d+21)) then
        tmp = t_1
    else if (u <= 5d+66) 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 = t1 * (v / (0.0 - (u * u)));
	double tmp;
	if (u <= -5.2e+21) {
		tmp = t_1;
	} else if (u <= 5e+66) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (v / (0.0 - (u * u)))
	tmp = 0
	if u <= -5.2e+21:
		tmp = t_1
	elif u <= 5e+66:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / Float64(0.0 - Float64(u * u))))
	tmp = 0.0
	if (u <= -5.2e+21)
		tmp = t_1;
	elseif (u <= 5e+66)
		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 = t1 * (v / (0.0 - (u * u)));
	tmp = 0.0;
	if (u <= -5.2e+21)
		tmp = t_1;
	elseif (u <= 5e+66)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(v / N[(0.0 - N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.2e+21], t$95$1, If[LessEqual[u, 5e+66], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.2e21 or 4.99999999999999991e66 < u

   1. Initial program 85.6%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 83.8%

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

   4. Applied egg-rr 83.8%

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

   5. Taylor expanded in t1 around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 81.9%

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

   7. Simplified 81.9%

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

   if -5.2e21 < u < 4.99999999999999991e66

   1. Initial program 65.0%

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

   3. Taylor expanded in t1 around 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 74.2%

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

   5. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 74.2%

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

   7. Applied egg-rr 74.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+21}:\\ \;\;\;\;t1 \cdot \frac{v}{0 - u \cdot u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+66}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{0 - u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{+121}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.5e+188)
   (/ (* v -0.5) u)
   (if (<= u 1.7e+121) (- 0.0 (/ v t1)) (/ -1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+188) {
		tmp = (v * -0.5) / u;
	} else if (u <= 1.7e+121) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = -1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-7.5d+188)) then
        tmp = (v * (-0.5d0)) / u
    else if (u <= 1.7d+121) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = (-1.0d0) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+188) {
		tmp = (v * -0.5) / u;
	} else if (u <= 1.7e+121) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = -1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.5e+188:
		tmp = (v * -0.5) / u
	elif u <= 1.7e+121:
		tmp = 0.0 - (v / t1)
	else:
		tmp = -1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.5e+188)
		tmp = Float64(Float64(v * -0.5) / u);
	elseif (u <= 1.7e+121)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = Float64(-1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.5e+188)
		tmp = (v * -0.5) / u;
	elseif (u <= 1.7e+121)
		tmp = 0.0 - (v / t1);
	else
		tmp = -1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.5e+188], N[(N[(v * -0.5), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 1.7e+121], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.4999999999999996e188

   1. Initial program 77.2%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \color{blue}{\left(2 \cdot \left(t1 \cdot u\right) + {u}^{2}\right)}\right) \]
   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 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 43.3%

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

   8. Simplified 43.3%

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

   if -7.4999999999999996e188 < u < 1.70000000000000005e121

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 64.8%

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

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 64.8%

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

   7. Applied egg-rr 64.8%

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

   if 1.70000000000000005e121 < u

   1. Initial program 86.2%

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

      1. *-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 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 95.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

      6. distribute-neg-frac N/A

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

      7. clear-num N/A

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

      12. /-lowering-/.f64 96.2%

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

   9. Applied egg-rr 96.2%

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

   10. Taylor expanded in t1 around inf

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

   12. Simplified 43.1%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{+121}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+187}:\\ \;\;\;\;\frac{v}{0 - u}\\ \mathbf{elif}\;u \leq 7.8 \cdot 10^{+120}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4e+187)
   (/ v (- 0.0 u))
   (if (<= u 7.8e+120) (- 0.0 (/ v t1)) (/ -1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+187) {
		tmp = v / (0.0 - u);
	} else if (u <= 7.8e+120) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = -1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4d+187)) then
        tmp = v / (0.0d0 - u)
    else if (u <= 7.8d+120) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = (-1.0d0) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+187) {
		tmp = v / (0.0 - u);
	} else if (u <= 7.8e+120) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = -1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4e+187:
		tmp = v / (0.0 - u)
	elif u <= 7.8e+120:
		tmp = 0.0 - (v / t1)
	else:
		tmp = -1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4e+187)
		tmp = Float64(v / Float64(0.0 - u));
	elseif (u <= 7.8e+120)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = Float64(-1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4e+187)
		tmp = v / (0.0 - u);
	elseif (u <= 7.8e+120)
		tmp = 0.0 - (v / t1);
	else
		tmp = -1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4e+187], N[(v / N[(0.0 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.8e+120], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.99999999999999963e187

   1. Initial program 77.2%

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

   3. Taylor expanded in t1 around inf

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

   5. Simplified 56.0%

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

   6. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 43.3%

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

   8. Simplified 43.3%

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

   if -3.99999999999999963e187 < u < 7.7999999999999997e120

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 64.8%

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

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 64.8%

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

   7. Applied egg-rr 64.8%

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

   if 7.7999999999999997e120 < u

   1. Initial program 86.2%

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

      1. *-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 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 95.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

      6. distribute-neg-frac N/A

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

      7. clear-num N/A

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

      12. /-lowering-/.f64 96.2%

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

   9. Applied egg-rr 96.2%

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

   10. Taylor expanded in t1 around inf

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

   12. Simplified 43.1%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+187}:\\ \;\;\;\;\frac{v}{0 - u}\\ \mathbf{elif}\;u \leq 7.8 \cdot 10^{+120}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{0 - u}\\ \mathbf{if}\;u \leq -6 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+121}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- 0.0 u))))
   (if (<= u -6e+186) t_1 (if (<= u 1.15e+121) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (0.0 - u);
	double tmp;
	if (u <= -6e+186) {
		tmp = t_1;
	} else if (u <= 1.15e+121) {
		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 / (0.0d0 - u)
    if (u <= (-6d+186)) then
        tmp = t_1
    else if (u <= 1.15d+121) 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 / (0.0 - u);
	double tmp;
	if (u <= -6e+186) {
		tmp = t_1;
	} else if (u <= 1.15e+121) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (0.0 - u)
	tmp = 0
	if u <= -6e+186:
		tmp = t_1
	elif u <= 1.15e+121:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(0.0 - u))
	tmp = 0.0
	if (u <= -6e+186)
		tmp = t_1;
	elseif (u <= 1.15e+121)
		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 / (0.0 - u);
	tmp = 0.0;
	if (u <= -6e+186)
		tmp = t_1;
	elseif (u <= 1.15e+121)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(0.0 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -6e+186], t$95$1, If[LessEqual[u, 1.15e+121], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.99999999999999964e186 or 1.1499999999999999e121 < u

   1. Initial program 83.2%

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

   3. Taylor expanded in t1 around inf

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

   5. Simplified 49.1%

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

   6. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 41.9%

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

   8. Simplified 41.9%

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

   if -5.99999999999999964e186 < u < 1.1499999999999999e121

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 64.8%

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

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 64.8%

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

   7. Applied egg-rr 64.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+186}:\\ \;\;\;\;\frac{v}{0 - u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+121}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% 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 73.4%

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

   1. *-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 97.6%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t1 around inf

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

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

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

   2. /-lowering-/.f64 97.6%

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

7. Simplified 97.6%

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

   1. /-lowering-/.f64 N/A

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

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

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

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

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. unsub-neg N/A

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

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

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

   8. /-lowering-/.f64 97.6%

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

9. Applied egg-rr 97.6%

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

Alternative 10: 62.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. *-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 97.6%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 60.1%

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

Alternative 11: 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 73.4%

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

   1. 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

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 97.8%

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

4. Applied egg-rr 97.8%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 59.8%

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

Alternative 12: 54.9% 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 73.4%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 51.1%

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

5. Simplified 51.1%

   \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
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 51.1%

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

7. Applied egg-rr 51.1%

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

8. Final simplification 51.1%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))