Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.8%

Time: 10.8s

Alternatives: 10

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: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. *-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

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

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

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

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

   9. neg-sub0 N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 99.1%

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

4. Applied egg-rr 99.1%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -2.55 \cdot 10^{-135}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(\left(0 - t1\right) - u\right)}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{0 - t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.8e+128)
     t_1
     (if (<= t1 -2.55e-135)
       (* t1 (/ v (* (+ t1 u) (- (- 0.0 t1) u))))
       (if (<= t1 1.4e+69) (* (/ v u) (/ (- 0.0 t1) u)) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.8e+128) {
		tmp = t_1;
	} else if (t1 <= -2.55e-135) {
		tmp = t1 * (v / ((t1 + u) * ((0.0 - t1) - u)));
	} else if (t1 <= 1.4e+69) {
		tmp = (v / u) * ((0.0 - 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 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.8d+128)) then
        tmp = t_1
    else if (t1 <= (-2.55d-135)) then
        tmp = t1 * (v / ((t1 + u) * ((0.0d0 - t1) - u)))
    else if (t1 <= 1.4d+69) then
        tmp = (v / u) * ((0.0d0 - 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 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.8e+128) {
		tmp = t_1;
	} else if (t1 <= -2.55e-135) {
		tmp = t1 * (v / ((t1 + u) * ((0.0 - t1) - u)));
	} else if (t1 <= 1.4e+69) {
		tmp = (v / u) * ((0.0 - t1) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.8e+128:
		tmp = t_1
	elif t1 <= -2.55e-135:
		tmp = t1 * (v / ((t1 + u) * ((0.0 - t1) - u)))
	elif t1 <= 1.4e+69:
		tmp = (v / u) * ((0.0 - t1) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.8e+128)
		tmp = t_1;
	elseif (t1 <= -2.55e-135)
		tmp = Float64(t1 * Float64(v / Float64(Float64(t1 + u) * Float64(Float64(0.0 - t1) - u))));
	elseif (t1 <= 1.4e+69)
		tmp = Float64(Float64(v / u) * Float64(Float64(0.0 - t1) / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.8e+128)
		tmp = t_1;
	elseif (t1 <= -2.55e-135)
		tmp = t1 * (v / ((t1 + u) * ((0.0 - t1) - u)));
	elseif (t1 <= 1.4e+69)
		tmp = (v / u) * ((0.0 - t1) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e+128], t$95$1, If[LessEqual[t1, -2.55e-135], N[(t1 * N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.4e+69], N[(N[(v / u), $MachinePrecision] * N[(N[(0.0 - t1), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.80000000000000014e128 or 1.39999999999999991e69 < t1

   1. Initial program 55.0%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

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

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

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

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

      9. neg-sub0 N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. frac-2neg N/A

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

      4. *-rgt-identity N/A

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

      5. sub0-neg N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in u around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 88.2%

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

   9. Simplified 88.2%

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

   if -1.80000000000000014e128 < t1 < -2.5500000000000001e-135

   1. Initial program 88.1%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 90.1%

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

   4. Applied egg-rr 90.1%

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

   if -2.5500000000000001e-135 < t1 < 1.39999999999999991e69

   1. Initial program 82.5%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 73.8%

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

   5. Simplified 73.8%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 83.2%

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -2.55 \cdot 10^{-135}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(\left(0 - t1\right) - u\right)}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{0 - t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{-76}:\\ \;\;\;\;0 - \frac{\frac{t1}{t1 + u}}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{-51}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.5e-76)
   (- 0.0 (/ (/ t1 (+ t1 u)) (/ u v)))
   (if (<= u 9e-51) (- 0.0 (/ v t1)) (- 0.0 (/ t1 (/ u (/ v u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.5e-76) {
		tmp = 0.0 - ((t1 / (t1 + u)) / (u / v));
	} else if (u <= 9e-51) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = 0.0 - (t1 / (u / (v / 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 (u <= (-3.5d-76)) then
        tmp = 0.0d0 - ((t1 / (t1 + u)) / (u / v))
    else if (u <= 9d-51) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = 0.0d0 - (t1 / (u / (v / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.5e-76) {
		tmp = 0.0 - ((t1 / (t1 + u)) / (u / v));
	} else if (u <= 9e-51) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = 0.0 - (t1 / (u / (v / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.5e-76:
		tmp = 0.0 - ((t1 / (t1 + u)) / (u / v))
	elif u <= 9e-51:
		tmp = 0.0 - (v / t1)
	else:
		tmp = 0.0 - (t1 / (u / (v / u)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.5e-76)
		tmp = Float64(0.0 - Float64(Float64(t1 / Float64(t1 + u)) / Float64(u / v)));
	elseif (u <= 9e-51)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = Float64(0.0 - Float64(t1 / Float64(u / Float64(v / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.5e-76)
		tmp = 0.0 - ((t1 / (t1 + u)) / (u / v));
	elseif (u <= 9e-51)
		tmp = 0.0 - (v / t1);
	else
		tmp = 0.0 - (t1 / (u / (v / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.5e-76], N[(0.0 - N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9e-51], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.49999999999999997e-76

   1. Initial program 77.4%

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

   3. Taylor expanded in t1 around 0

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

   5. Simplified 71.0%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. distribute-frac-neg N/A

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 79.4%

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

   7. Applied egg-rr 79.4%

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

   if -3.49999999999999997e-76 < u < 8.99999999999999948e-51

   1. Initial program 62.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 84.2%

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

   5. Simplified 84.2%

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

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 84.2%

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

   7. Applied egg-rr 84.2%

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

   if 8.99999999999999948e-51 < u

   1. Initial program 83.8%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 73.5%

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

   5. Simplified 73.5%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. /-lowering-/.f64 82.8%

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

   9. Applied egg-rr 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{-76}:\\ \;\;\;\;0 - \frac{\frac{t1}{t1 + u}}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{-51}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{0 - v}}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{-51}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.9e-75)
   (/ (/ t1 u) (/ u (- 0.0 v)))
   (if (<= u 2.05e-51) (- 0.0 (/ v t1)) (- 0.0 (/ t1 (/ u (/ v u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-75) {
		tmp = (t1 / u) / (u / (0.0 - v));
	} else if (u <= 2.05e-51) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = 0.0 - (t1 / (u / (v / 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 (u <= (-2.9d-75)) then
        tmp = (t1 / u) / (u / (0.0d0 - v))
    else if (u <= 2.05d-51) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = 0.0d0 - (t1 / (u / (v / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-75) {
		tmp = (t1 / u) / (u / (0.0 - v));
	} else if (u <= 2.05e-51) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = 0.0 - (t1 / (u / (v / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.9e-75:
		tmp = (t1 / u) / (u / (0.0 - v))
	elif u <= 2.05e-51:
		tmp = 0.0 - (v / t1)
	else:
		tmp = 0.0 - (t1 / (u / (v / u)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.9e-75)
		tmp = Float64(Float64(t1 / u) / Float64(u / Float64(0.0 - v)));
	elseif (u <= 2.05e-51)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = Float64(0.0 - Float64(t1 / Float64(u / Float64(v / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.9e-75)
		tmp = (t1 / u) / (u / (0.0 - v));
	elseif (u <= 2.05e-51)
		tmp = 0.0 - (v / t1);
	else
		tmp = 0.0 - (t1 / (u / (v / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.9e-75], N[(N[(t1 / u), $MachinePrecision] / N[(u / N[(0.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.05e-51], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.9000000000000002e-75

   1. Initial program 77.4%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.1%

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

   5. Simplified 67.1%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 74.9%

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

   7. Applied egg-rr 74.9%

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

      1. clear-num N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. /-lowering-/.f64 79.4%

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

   9. Applied egg-rr 79.4%

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

   if -2.9000000000000002e-75 < u < 2.04999999999999987e-51

   1. Initial program 62.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 84.2%

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

   5. Simplified 84.2%

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

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 84.2%

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

   7. Applied egg-rr 84.2%

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

   if 2.04999999999999987e-51 < u

   1. Initial program 83.8%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 73.5%

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

   5. Simplified 73.5%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. /-lowering-/.f64 82.8%

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

   9. Applied egg-rr 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{0 - v}}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{-51}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.95 \cdot 10^{-75}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{0 - t1}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-51}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.95e-75)
   (* (/ v u) (/ (- 0.0 t1) u))
   (if (<= u 3.3e-51) (- 0.0 (/ v t1)) (- 0.0 (/ t1 (/ u (/ v u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.95e-75) {
		tmp = (v / u) * ((0.0 - t1) / u);
	} else if (u <= 3.3e-51) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = 0.0 - (t1 / (u / (v / 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 (u <= (-2.95d-75)) then
        tmp = (v / u) * ((0.0d0 - t1) / u)
    else if (u <= 3.3d-51) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = 0.0d0 - (t1 / (u / (v / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.95e-75) {
		tmp = (v / u) * ((0.0 - t1) / u);
	} else if (u <= 3.3e-51) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = 0.0 - (t1 / (u / (v / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.95e-75:
		tmp = (v / u) * ((0.0 - t1) / u)
	elif u <= 3.3e-51:
		tmp = 0.0 - (v / t1)
	else:
		tmp = 0.0 - (t1 / (u / (v / u)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.95e-75)
		tmp = Float64(Float64(v / u) * Float64(Float64(0.0 - t1) / u));
	elseif (u <= 3.3e-51)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = Float64(0.0 - Float64(t1 / Float64(u / Float64(v / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.95e-75)
		tmp = (v / u) * ((0.0 - t1) / u);
	elseif (u <= 3.3e-51)
		tmp = 0.0 - (v / t1);
	else
		tmp = 0.0 - (t1 / (u / (v / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.95e-75], N[(N[(v / u), $MachinePrecision] * N[(N[(0.0 - t1), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.3e-51], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.95e-75

   1. Initial program 77.4%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.1%

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

   5. Simplified 67.1%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 74.9%

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

   7. Applied egg-rr 74.9%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 78.3%

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

   9. Applied egg-rr 78.3%

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

   if -2.95e-75 < u < 3.29999999999999973e-51

   1. Initial program 62.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 84.2%

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

   5. Simplified 84.2%

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

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 84.2%

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

   7. Applied egg-rr 84.2%

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

   if 3.29999999999999973e-51 < u

   1. Initial program 83.8%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 73.5%

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

   5. Simplified 73.5%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. /-lowering-/.f64 82.8%

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

   9. Applied egg-rr 82.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.95 \cdot 10^{-75}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{0 - t1}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-51}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u} \cdot \frac{0 - t1}{u}\\ \mathbf{if}\;u \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{-78}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v u) (/ (- 0.0 t1) u))))
   (if (<= u -3.6e-75) t_1 (if (<= u 5.6e-78) (- 0.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) * ((0.0 - t1) / u);
	double tmp;
	if (u <= -3.6e-75) {
		tmp = t_1;
	} else if (u <= 5.6e-78) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / u) * ((0.0d0 - t1) / u)
    if (u <= (-3.6d-75)) then
        tmp = t_1
    else if (u <= 5.6d-78) then
        tmp = 0.0d0 - (v / t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / u) * ((0.0 - t1) / u);
	double tmp;
	if (u <= -3.6e-75) {
		tmp = t_1;
	} else if (u <= 5.6e-78) {
		tmp = 0.0 - (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) * ((0.0 - t1) / u)
	tmp = 0
	if u <= -3.6e-75:
		tmp = t_1
	elif u <= 5.6e-78:
		tmp = 0.0 - (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) * Float64(Float64(0.0 - t1) / u))
	tmp = 0.0
	if (u <= -3.6e-75)
		tmp = t_1;
	elseif (u <= 5.6e-78)
		tmp = Float64(0.0 - Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / u) * ((0.0 - t1) / u);
	tmp = 0.0;
	if (u <= -3.6e-75)
		tmp = t_1;
	elseif (u <= 5.6e-78)
		tmp = 0.0 - (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / u), $MachinePrecision] * N[(N[(0.0 - t1), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.6e-75], t$95$1, If[LessEqual[u, 5.6e-78], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.6e-75 or 5.60000000000000047e-78 < u

   1. Initial program 80.2%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 70.0%

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

   5. Simplified 70.0%

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

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. remove-double-neg N/A

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-lowering-*.f64 76.2%

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

   7. Applied egg-rr 76.2%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 79.6%

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

   9. Applied egg-rr 79.6%

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

   if -3.6e-75 < u < 5.60000000000000047e-78

   1. Initial program 62.3%

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

   3. Taylor expanded in t1 around 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 86.0%

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

   5. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 86.0%

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

   7. Applied egg-rr 86.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{0 - t1}{u}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{-78}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{0 - t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{u}\\ \mathbf{if}\;u \leq -1.42 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v u))))
   (if (<= u -1.42e+156) t_1 (if (<= u 1.5e+136) (- 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 <= -1.42e+156) {
		tmp = t_1;
	} else if (u <= 1.5e+136) {
		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 <= (-1.42d+156)) then
        tmp = t_1
    else if (u <= 1.5d+136) 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 <= -1.42e+156) {
		tmp = t_1;
	} else if (u <= 1.5e+136) {
		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 <= -1.42e+156:
		tmp = t_1
	elif u <= 1.5e+136:
		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 <= -1.42e+156)
		tmp = t_1;
	elseif (u <= 1.5e+136)
		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 <= -1.42e+156)
		tmp = t_1;
	elseif (u <= 1.5e+136)
		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, -1.42e+156], t$95$1, If[LessEqual[u, 1.5e+136], N[(0.0 - N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.41999999999999998e156 or 1.49999999999999989e136 < u

   1. Initial program 76.7%

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

   3. Taylor expanded in t1 around 0

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

   5. Simplified 76.7%

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

   6. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 41.6%

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

   8. Simplified 41.6%

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

   if -1.41999999999999998e156 < u < 1.49999999999999989e136

   1. Initial program 73.4%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 60.8%

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

   5. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
   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 60.8%

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

   7. Applied egg-rr 60.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.42 \cdot 10^{+156}:\\ \;\;\;\;0 - \frac{v}{u}\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;0 - \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / ((u * (-2.0d0)) - t1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. *-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

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

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

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

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

   9. neg-sub0 N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 99.1%

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

4. Applied egg-rr 99.1%

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

   1. clear-num N/A

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

   2. frac-times N/A

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

   3. frac-2neg N/A

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

   4. *-rgt-identity N/A

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

   5. sub0-neg N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

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

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

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

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

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

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

   14. +-lowering-+.f64 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 56.6%

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

9. Simplified 56.6%

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

Alternative 9: 61.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. +-lowering-+.f64 71.8%

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

4. Applied egg-rr 71.8%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 42.0%

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

8. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. +-lowering-+.f64 56.4%

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

10. Simplified 56.4%

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

    1. sub0-neg N/A

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

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

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

    3. +-commutative N/A

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

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

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

    5. +-lowering-+.f64 56.4%

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

12. Applied egg-rr 56.4%

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

13. Final simplification 56.4%

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

Alternative 10: 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 74.6%

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

3. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. --lowering--.f64 47.0%

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

5. Simplified 47.0%

   \[\leadsto \color{blue}{\frac{v}{0 - t1}} \]
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 47.0%

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

7. Applied egg-rr 47.0%

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

8. Final simplification 47.0%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))