Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.5%

Time: 10.7s

Alternatives: 9

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: 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: 97.5% 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 72.5%

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

   1. 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 98.5%

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

4. Applied egg-rr 98.5%

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

   1. sub0-neg N/A

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

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

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

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

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

   4. +-lowering-+.f64 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0 - v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.65 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8 \cdot 10^{-87}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- 0.0 v) (+ t1 u))))
   (if (<= t1 -1.65e+19)
     t_1
     (if (<= t1 8e-87) (/ (- 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 <= -1.65e+19) {
		tmp = t_1;
	} else if (t1 <= 8e-87) {
		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 <= (-1.65d+19)) then
        tmp = t_1
    else if (t1 <= 8d-87) 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 <= -1.65e+19) {
		tmp = t_1;
	} else if (t1 <= 8e-87) {
		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 <= -1.65e+19:
		tmp = t_1
	elif t1 <= 8e-87:
		tmp = (0.0 - (t1 / u)) / (u / v)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(0.0 - v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.65e+19)
		tmp = t_1;
	elseif (t1 <= 8e-87)
		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 <= -1.65e+19)
		tmp = t_1;
	elseif (t1 <= 8e-87)
		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[(0.0 - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.65e+19], t$95$1, If[LessEqual[t1, 8e-87], 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 < -1.65e19 or 8.00000000000000014e-87 < t1

   1. Initial program 67.2%

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

      1. 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 98.9%

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

   4. Applied egg-rr 98.9%

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

   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 84.8%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. +-lowering-+.f64 85.2%

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

   9. Applied egg-rr 85.2%

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

   if -1.65e19 < t1 < 8.00000000000000014e-87

   1. Initial program 78.3%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 62.7%

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

   5. Simplified 62.7%

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

      1. frac-2neg N/A

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

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

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. times-frac N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. frac-2neg N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 76.4%

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

   9. Applied egg-rr 76.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{0 - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8 \cdot 10^{-87}:\\ \;\;\;\;\frac{0 - \frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0 - v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-85}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(0 - \frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- 0.0 v) (+ t1 u))))
   (if (<= t1 -1.22e+27)
     t_1
     (if (<= t1 3e-85) (* (/ t1 u) (- 0.0 (/ v u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (0.0 - v) / (t1 + u);
	double tmp;
	if (t1 <= -1.22e+27) {
		tmp = t_1;
	} else if (t1 <= 3e-85) {
		tmp = (t1 / u) * (0.0 - (v / 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 <= (-1.22d+27)) then
        tmp = t_1
    else if (t1 <= 3d-85) then
        tmp = (t1 / u) * (0.0d0 - (v / 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 <= -1.22e+27) {
		tmp = t_1;
	} else if (t1 <= 3e-85) {
		tmp = (t1 / u) * (0.0 - (v / u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (0.0 - v) / (t1 + u)
	tmp = 0
	if t1 <= -1.22e+27:
		tmp = t_1
	elif t1 <= 3e-85:
		tmp = (t1 / u) * (0.0 - (v / u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(0.0 - v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.22e+27)
		tmp = t_1;
	elseif (t1 <= 3e-85)
		tmp = Float64(Float64(t1 / u) * Float64(0.0 - Float64(v / 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 <= -1.22e+27)
		tmp = t_1;
	elseif (t1 <= 3e-85)
		tmp = (t1 / u) * (0.0 - (v / u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(0.0 - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.22e+27], t$95$1, If[LessEqual[t1, 3e-85], N[(N[(t1 / u), $MachinePrecision] * N[(0.0 - N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.2200000000000001e27 or 3.00000000000000022e-85 < t1

   1. Initial program 67.2%

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

      1. 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 98.9%

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

   4. Applied egg-rr 98.9%

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

   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 84.8%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. +-lowering-+.f64 85.2%

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

   9. Applied egg-rr 85.2%

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

   if -1.2200000000000001e27 < t1 < 3.00000000000000022e-85

   1. Initial program 78.3%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 62.7%

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

   5. Simplified 62.7%

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

      1. frac-2neg N/A

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

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

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

      3. remove-double-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. times-frac N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 76.1%

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

   7. Applied egg-rr 76.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+27}:\\ \;\;\;\;\frac{0 - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-85}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(0 - \frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -3.2 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.26 \cdot 10^{-33}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (* t1 (/ v (* u u))))))
   (if (<= u -3.2e-44) t_1 (if (<= u 1.26e-33) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (t1 * (v / (u * u)));
	double tmp;
	if (u <= -3.2e-44) {
		tmp = t_1;
	} else if (u <= 1.26e-33) {
		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 = 0.0d0 - (t1 * (v / (u * u)))
    if (u <= (-3.2d-44)) then
        tmp = t_1
    else if (u <= 1.26d-33) 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 = 0.0 - (t1 * (v / (u * u)));
	double tmp;
	if (u <= -3.2e-44) {
		tmp = t_1;
	} else if (u <= 1.26e-33) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (t1 * (v / (u * u)))
	tmp = 0
	if u <= -3.2e-44:
		tmp = t_1
	elif u <= 1.26e-33:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(t1 * Float64(v / Float64(u * u))))
	tmp = 0.0
	if (u <= -3.2e-44)
		tmp = t_1;
	elseif (u <= 1.26e-33)
		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 = 0.0 - (t1 * (v / (u * u)));
	tmp = 0.0;
	if (u <= -3.2e-44)
		tmp = t_1;
	elseif (u <= 1.26e-33)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.2e-44], t$95$1, If[LessEqual[u, 1.26e-33], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.19999999999999995e-44 or 1.26000000000000005e-33 < u

   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

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 70.1%

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

   5. Simplified 70.1%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 73.3%

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

   7. Applied egg-rr 73.3%

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

   if -3.19999999999999995e-44 < u < 1.26000000000000005e-33

   1. Initial program 63.0%

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

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

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

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

      4. /-lowering-/.f64 80.7%

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

   5. Simplified 80.7%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 80.7%

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

   7. Applied egg-rr 80.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{-44}:\\ \;\;\;\;0 - t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.26 \cdot 10^{-33}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+198}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{t1 + u} \cdot \left(0 - t1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e+198)
   (- 0.0 (/ v t1))
   (* (/ (/ v (+ t1 u)) (+ t1 u)) (- 0.0 t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+198) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = ((v / (t1 + u)) / (t1 + u)) * (0.0 - 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 <= (-1.2d+198)) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = ((v / (t1 + u)) / (t1 + u)) * (0.0d0 - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+198) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = ((v / (t1 + u)) / (t1 + u)) * (0.0 - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e+198:
		tmp = 0.0 - (v / t1)
	else:
		tmp = ((v / (t1 + u)) / (t1 + u)) * (0.0 - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e+198)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = Float64(Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)) * Float64(0.0 - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e+198)
		tmp = 0.0 - (v / t1);
	else
		tmp = ((v / (t1 + u)) / (t1 + u)) * (0.0 - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e+198], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.2000000000000001e198

   1. Initial program 34.2%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 96.6%

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

   5. Simplified 96.6%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 96.6%

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

   7. Applied egg-rr 96.6%

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

   if -1.2000000000000001e198 < t1

   1. Initial program 76.9%

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

      1. 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. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 92.6%

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

   4. Applied egg-rr 92.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+198}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{t1 + u} \cdot \left(0 - t1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\frac{u}{v}}\\ \mathbf{if}\;u \leq -1.25 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+179}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ -1.0 (/ u v))))
   (if (<= u -1.25e+181) t_1 (if (<= u 6.2e+179) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 / (u / v);
	double tmp;
	if (u <= -1.25e+181) {
		tmp = t_1;
	} else if (u <= 6.2e+179) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-1.0d0) / (u / v)
    if (u <= (-1.25d+181)) then
        tmp = t_1
    else if (u <= 6.2d+179) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -1.0 / (u / v);
	double tmp;
	if (u <= -1.25e+181) {
		tmp = t_1;
	} else if (u <= 6.2e+179) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 / (u / v)
	tmp = 0
	if u <= -1.25e+181:
		tmp = t_1
	elif u <= 6.2e+179:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 / Float64(u / v))
	tmp = 0.0
	if (u <= -1.25e+181)
		tmp = t_1;
	elseif (u <= 6.2e+179)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 / (u / v);
	tmp = 0.0;
	if (u <= -1.25e+181)
		tmp = t_1;
	elseif (u <= 6.2e+179)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.25e+181], t$95$1, If[LessEqual[u, 6.2e+179], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.2500000000000001e181 or 6.2e179 < u

   1. Initial program 81.3%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      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 99.3%

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

   4. Applied egg-rr 99.3%

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

   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 43.6%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 41.7%

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

   if -1.2500000000000001e181 < u < 6.2e179

   1. Initial program 70.6%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 63.1%

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

   5. Simplified 63.1%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 63.1%

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

   7. Applied egg-rr 63.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+181}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+179}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{u}\\ \mathbf{if}\;u \leq -1.7 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.05 \cdot 10^{+178}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v u))))
   (if (<= u -1.7e+181) t_1 (if (<= u 3.05e+178) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / u);
	double tmp;
	if (u <= -1.7e+181) {
		tmp = t_1;
	} else if (u <= 3.05e+178) {
		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 = 0.0d0 - (v / u)
    if (u <= (-1.7d+181)) then
        tmp = t_1
    else if (u <= 3.05d+178) 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 = 0.0 - (v / u);
	double tmp;
	if (u <= -1.7e+181) {
		tmp = t_1;
	} else if (u <= 3.05e+178) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (v / u)
	tmp = 0
	if u <= -1.7e+181:
		tmp = t_1
	elif u <= 3.05e+178:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(v / u))
	tmp = 0.0
	if (u <= -1.7e+181)
		tmp = t_1;
	elseif (u <= 3.05e+178)
		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 = 0.0 - (v / u);
	tmp = 0.0;
	if (u <= -1.7e+181)
		tmp = t_1;
	elseif (u <= 3.05e+178)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.7e+181], t$95$1, If[LessEqual[u, 3.05e+178], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.70000000000000015e181 or 3.05e178 < u

   1. Initial program 81.3%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      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 99.3%

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

   4. Applied egg-rr 99.3%

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

   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 43.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. +-lowering-+.f64 43.5%

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

   9. Applied egg-rr 43.5%

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

   10. Taylor expanded in t1 around 0

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

       1. /-lowering-/.f64 41.6%

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

   12. Simplified 41.6%

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

   if -1.70000000000000015e181 < u < 3.05e178

   1. Initial program 70.6%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 63.1%

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

   5. Simplified 63.1%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 63.1%

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

   7. Applied egg-rr 63.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+181}:\\ \;\;\;\;0 - \frac{v}{u}\\ \mathbf{elif}\;u \leq 3.05 \cdot 10^{+178}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 1.7× speedup

Math

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

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(Float64(0.0 - 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[(N[(0.0 - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. 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 98.5%

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

4. Applied egg-rr 98.5%

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

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 59.6%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. mul-1-neg N/A

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

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

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

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

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

   6. +-lowering-+.f64 59.9%

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

9. Applied egg-rr 59.9%

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

10. Final simplification 59.9%

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

Alternative 9: 53.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. /-lowering-/.f64 54.2%

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

5. Simplified 54.2%

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

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 54.2%

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

7. Applied egg-rr 54.2%

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

8. Final simplification 54.2%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))