Rosa's DopplerBench

Percentage Accurate: 73.0% → 97.9%

Time: 10.7s

Alternatives: 7

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% 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: 97.9% 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 75.0%

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

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

   9. remove-double-neg N/A

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

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

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

   11. neg-sub0 N/A

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

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

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

   13. +-lowering-+.f64 98.0%

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

4. Applied egg-rr 98.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

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

   4. sub0-neg N/A

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

   5. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. +-lowering-+.f64 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 2: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{0 - t1}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -8.2e-165)
   (/ (/ t1 (/ (+ t1 u) v)) (- 0.0 t1))
   (if (<= t1 9.5e+65) (/ (- 0.0 (/ t1 u)) (/ u v)) (- 0.0 (/ v (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8.2e-165) {
		tmp = (t1 / ((t1 + u) / v)) / (0.0 - t1);
	} else if (t1 <= 9.5e+65) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = 0.0 - (v / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-8.2d-165)) then
        tmp = (t1 / ((t1 + u) / v)) / (0.0d0 - t1)
    else if (t1 <= 9.5d+65) then
        tmp = (0.0d0 - (t1 / u)) / (u / v)
    else
        tmp = 0.0d0 - (v / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8.2e-165) {
		tmp = (t1 / ((t1 + u) / v)) / (0.0 - t1);
	} else if (t1 <= 9.5e+65) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = 0.0 - (v / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -8.2e-165:
		tmp = (t1 / ((t1 + u) / v)) / (0.0 - t1)
	elif t1 <= 9.5e+65:
		tmp = (0.0 - (t1 / u)) / (u / v)
	else:
		tmp = 0.0 - (v / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -8.2e-165)
		tmp = Float64(Float64(t1 / Float64(Float64(t1 + u) / v)) / Float64(0.0 - t1));
	elseif (t1 <= 9.5e+65)
		tmp = Float64(Float64(0.0 - Float64(t1 / u)) / Float64(u / v));
	else
		tmp = Float64(0.0 - Float64(v / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -8.2e-165)
		tmp = (t1 / ((t1 + u) / v)) / (0.0 - t1);
	elseif (t1 <= 9.5e+65)
		tmp = (0.0 - (t1 / u)) / (u / v);
	else
		tmp = 0.0 - (v / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -8.2e-165], N[(N[(t1 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 9.5e+65], N[(N[(0.0 - N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.2000000000000004e-165

   1. Initial program 70.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.5%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

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

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. distribute-frac-neg N/A

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

   if -8.2000000000000004e-165 < t1 < 9.5000000000000005e65

   1. Initial program 82.2%

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

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 79.9%

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

   5. Simplified 79.9%

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

      1. sub0-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. distribute-frac-neg N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 85.8%

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

   9. Applied egg-rr 85.8%

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

   if 9.5000000000000005e65 < t1

   1. Initial program 71.3%

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

      1. *-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. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

      13. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. sub0-neg N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. +-lowering-+.f64 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in t1 around inf

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

   11. Simplified 87.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{0 - t1}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -4.3 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (+ t1 u)))))
   (if (<= t1 -4.3e-138)
     t_1
     (if (<= t1 9.5e+65) (/ (- 0.0 (/ t1 u)) (/ u v)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -4.3e-138) {
		tmp = t_1;
	} else if (t1 <= 9.5e+65) {
		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 = 0.0d0 - (v / (t1 + u))
    if (t1 <= (-4.3d-138)) then
        tmp = t_1
    else if (t1 <= 9.5d+65) 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 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -4.3e-138) {
		tmp = t_1;
	} else if (t1 <= 9.5e+65) {
		tmp = (0.0 - (t1 / u)) / (u / v);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (v / (t1 + u))
	tmp = 0
	if t1 <= -4.3e-138:
		tmp = t_1
	elif t1 <= 9.5e+65:
		tmp = (0.0 - (t1 / u)) / (u / v)
	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 <= -4.3e-138)
		tmp = t_1;
	elseif (t1 <= 9.5e+65)
		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 = 0.0 - (v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -4.3e-138)
		tmp = t_1;
	elseif (t1 <= 9.5e+65)
		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[(0.0 - N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.3e-138], t$95$1, If[LessEqual[t1, 9.5e+65], 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 < -4.3e-138 or 9.5000000000000005e65 < t1

   1. Initial program 70.9%

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

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

      9. remove-double-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

      13. +-lowering-+.f64 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. sub0-neg N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. +-lowering-+.f64 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in t1 around inf

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

   11. Simplified 83.7%

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

   if -4.3e-138 < t1 < 9.5000000000000005e65

   1. Initial program 81.6%

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

   3. Taylor expanded in t1 around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 79.0%

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

   5. Simplified 79.0%

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

      1. sub0-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. distribute-frac-neg N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 84.6%

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

   9. Applied egg-rr 84.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.3 \cdot 10^{-138}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{+65}:\\ \;\;\;\;0 - \frac{v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (+ t1 u)))))
   (if (<= t1 -3.8e-138)
     t_1
     (if (<= t1 9.5e+65) (- 0.0 (/ v (/ 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 <= -3.8e-138) {
		tmp = t_1;
	} else if (t1 <= 9.5e+65) {
		tmp = 0.0 - (v / (u / (t1 / u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0d0 - (v / (t1 + u))
    if (t1 <= (-3.8d-138)) then
        tmp = t_1
    else if (t1 <= 9.5d+65) then
        tmp = 0.0d0 - (v / (u / (t1 / u)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -3.8e-138) {
		tmp = t_1;
	} else if (t1 <= 9.5e+65) {
		tmp = 0.0 - (v / (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 <= -3.8e-138:
		tmp = t_1
	elif t1 <= 9.5e+65:
		tmp = 0.0 - (v / (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 <= -3.8e-138)
		tmp = t_1;
	elseif (t1 <= 9.5e+65)
		tmp = Float64(0.0 - Float64(v / 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 <= -3.8e-138)
		tmp = t_1;
	elseif (t1 <= 9.5e+65)
		tmp = 0.0 - (v / (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, -3.8e-138], t$95$1, If[LessEqual[t1, 9.5e+65], N[(0.0 - N[(v / N[(u / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.8000000000000002e-138 or 9.5000000000000005e65 < t1

   1. Initial program 70.9%

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

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

      9. remove-double-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

      13. +-lowering-+.f64 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. sub0-neg N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. +-lowering-+.f64 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in t1 around inf

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

   11. Simplified 83.7%

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

   if -3.8000000000000002e-138 < t1 < 9.5000000000000005e65

   1. Initial program 81.6%

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

   3. Taylor expanded in t1 around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 79.0%

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

   5. Simplified 79.0%

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

      1. sub0-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. associate-*l/ N/A

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

      2. distribute-neg-frac2 N/A

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

      3. clear-num N/A

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

      4. sub0-neg N/A

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

      5. div-inv N/A

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

      6. associate-/l/ N/A

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

      7. frac-2neg N/A

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

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

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

      9. sub0-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. frac-2neg N/A

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

      12. distribute-frac-neg N/A

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

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

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

      14. frac-2neg N/A

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

      15. distribute-neg-frac2 N/A

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

      16. sub0-neg N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. div-inv N/A

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

      21. sub0-neg N/A

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

   9. Applied egg-rr 79.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-138}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{+65}:\\ \;\;\;\;0 - \frac{v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-1}{u}\\ \mathbf{if}\;u \leq -1.6 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{+234}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ -1.0 u))))
   (if (<= u -1.6e+175) t_1 (if (<= u 2.1e+234) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-1.0 / u);
	double tmp;
	if (u <= -1.6e+175) {
		tmp = t_1;
	} else if (u <= 2.1e+234) {
		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 * ((-1.0d0) / u)
    if (u <= (-1.6d+175)) then
        tmp = t_1
    else if (u <= 2.1d+234) 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 * (-1.0 / u);
	double tmp;
	if (u <= -1.6e+175) {
		tmp = t_1;
	} else if (u <= 2.1e+234) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-1.0 / u)
	tmp = 0
	if u <= -1.6e+175:
		tmp = t_1
	elif u <= 2.1e+234:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(-1.0 / u))
	tmp = 0.0
	if (u <= -1.6e+175)
		tmp = t_1;
	elseif (u <= 2.1e+234)
		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 * (-1.0 / u);
	tmp = 0.0;
	if (u <= -1.6e+175)
		tmp = t_1;
	elseif (u <= 2.1e+234)
		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[(-1.0 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.6e+175], t$95$1, If[LessEqual[u, 2.1e+234], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.60000000000000011e175 or 2.1e234 < u

   1. Initial program 81.5%

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

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

      9. remove-double-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

      13. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. /-lowering-/.f64 93.8%

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

   7. Simplified 93.8%

      \[\leadsto \frac{\color{blue}{\frac{v}{u}}}{\frac{0 - \left(t1 + u\right)}{t1}} \]
   8. Step-by-step derivation

      1. div-inv N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. inv-pow N/A

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

      6. inv-pow N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. inv-pow N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. distribute-neg-frac2 N/A

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

      17. distribute-frac-neg N/A

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

      18. sub0-neg N/A

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

      19. frac-2neg N/A

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

      20. /-lowering-/.f64 N/A

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

      21. /-lowering-/.f64 N/A

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

      22. +-lowering-+.f64 84.5%

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

   9. Applied egg-rr 84.5%

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

   10. Taylor expanded in u around 0

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

       1. /-lowering-/.f64 59.0%

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

   12. Simplified 59.0%

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

   if -1.60000000000000011e175 < u < 2.1e234

   1. Initial program 73.8%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 62.8%

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

   5. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 62.8%

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

   7. Applied egg-rr 62.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+175}:\\ \;\;\;\;v \cdot \frac{-1}{u}\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{+234}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.2% 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 75.0%

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

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

   9. remove-double-neg N/A

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

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

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

   11. neg-sub0 N/A

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

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

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

   13. +-lowering-+.f64 98.0%

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

4. Applied egg-rr 98.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

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

   4. sub0-neg N/A

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

   5. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. +-lowering-+.f64 98.1%

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

6. Applied egg-rr 98.1%

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

   9. +-lowering-+.f64 97.8%

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

8. Applied egg-rr 97.8%

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

9. Taylor expanded in t1 around inf

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

11. Simplified 64.5%

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

12. Final simplification 64.5%

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

Alternative 7: 53.6% 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 75.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 56.0%

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

5. Simplified 56.0%

   \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
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 56.0%

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

7. Applied egg-rr 56.0%

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

8. Final simplification 56.0%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))