Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.1%

Time: 11.4s

Alternatives: 14

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: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (t1 * (v / (t1 + u))) / ((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 = (t1 * (v / (t1 + u))) / ((0.0d0 - t1) - u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. distribute-neg-frac2 N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

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

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

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

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

   10. +-commutative N/A

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

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

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

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

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

   13. +-commutative N/A

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

   14. +-lowering-+.f64 98.8%

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

5. Simplified 98.8%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

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

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

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

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

   7. +-lowering-+.f64 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0 - t1\right) - u\\ t_2 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{u \cdot 2}{\frac{t1}{v}} - v}{t1}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{+145}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- 0.0 t1) u)) (t_2 (* v (/ t1 (* (+ t1 u) t_1)))))
   (if (<= t1 -5.4e+171)
     (/ (- (/ (* u 2.0) (/ t1 v)) v) t1)
     (if (<= t1 -3.5e-149)
       t_2
       (if (<= t1 2.4e-128)
         (/ -1.0 (/ u (/ t1 (/ u v))))
         (if (<= t1 1.25e+145) t_2 (/ v t_1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (0.0 - t1) - u;
	double t_2 = v * (t1 / ((t1 + u) * t_1));
	double tmp;
	if (t1 <= -5.4e+171) {
		tmp = (((u * 2.0) / (t1 / v)) - v) / t1;
	} else if (t1 <= -3.5e-149) {
		tmp = t_2;
	} else if (t1 <= 2.4e-128) {
		tmp = -1.0 / (u / (t1 / (u / v)));
	} else if (t1 <= 1.25e+145) {
		tmp = t_2;
	} else {
		tmp = v / 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) :: t_2
    real(8) :: tmp
    t_1 = (0.0d0 - t1) - u
    t_2 = v * (t1 / ((t1 + u) * t_1))
    if (t1 <= (-5.4d+171)) then
        tmp = (((u * 2.0d0) / (t1 / v)) - v) / t1
    else if (t1 <= (-3.5d-149)) then
        tmp = t_2
    else if (t1 <= 2.4d-128) then
        tmp = (-1.0d0) / (u / (t1 / (u / v)))
    else if (t1 <= 1.25d+145) then
        tmp = t_2
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (0.0 - t1) - u;
	double t_2 = v * (t1 / ((t1 + u) * t_1));
	double tmp;
	if (t1 <= -5.4e+171) {
		tmp = (((u * 2.0) / (t1 / v)) - v) / t1;
	} else if (t1 <= -3.5e-149) {
		tmp = t_2;
	} else if (t1 <= 2.4e-128) {
		tmp = -1.0 / (u / (t1 / (u / v)));
	} else if (t1 <= 1.25e+145) {
		tmp = t_2;
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (0.0 - t1) - u
	t_2 = v * (t1 / ((t1 + u) * t_1))
	tmp = 0
	if t1 <= -5.4e+171:
		tmp = (((u * 2.0) / (t1 / v)) - v) / t1
	elif t1 <= -3.5e-149:
		tmp = t_2
	elif t1 <= 2.4e-128:
		tmp = -1.0 / (u / (t1 / (u / v)))
	elif t1 <= 1.25e+145:
		tmp = t_2
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(0.0 - t1) - u)
	t_2 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)))
	tmp = 0.0
	if (t1 <= -5.4e+171)
		tmp = Float64(Float64(Float64(Float64(u * 2.0) / Float64(t1 / v)) - v) / t1);
	elseif (t1 <= -3.5e-149)
		tmp = t_2;
	elseif (t1 <= 2.4e-128)
		tmp = Float64(-1.0 / Float64(u / Float64(t1 / Float64(u / v))));
	elseif (t1 <= 1.25e+145)
		tmp = t_2;
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (0.0 - t1) - u;
	t_2 = v * (t1 / ((t1 + u) * t_1));
	tmp = 0.0;
	if (t1 <= -5.4e+171)
		tmp = (((u * 2.0) / (t1 / v)) - v) / t1;
	elseif (t1 <= -3.5e-149)
		tmp = t_2;
	elseif (t1 <= 2.4e-128)
		tmp = -1.0 / (u / (t1 / (u / v)));
	elseif (t1 <= 1.25e+145)
		tmp = t_2;
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.4e+171], N[(N[(N[(N[(u * 2.0), $MachinePrecision] / N[(t1 / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, -3.5e-149], t$95$2, If[LessEqual[t1, 2.4e-128], N[(-1.0 / N[(u / N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.25e+145], t$95$2, N[(v / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -5.3999999999999996e171

   1. Initial program 24.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

      12. +-lowering-+.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 78.2%

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

   7. Simplified 78.2%

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

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

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(u \cdot 2\right) \cdot \frac{1}{\frac{t1}{v}}\right), v\right), t1\right) \]

      6. un-div-inv N/A

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

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

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

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

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

      9. /-lowering-/.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -5.3999999999999996e171 < t1 < -3.5e-149 or 2.3999999999999998e-128 < t1 < 1.24999999999999992e145

   1. Initial program 83.9%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 99.9%

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

   5. Simplified 99.9%

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

   7. Applied egg-rr 90.8%

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

   if -3.5e-149 < t1 < 2.3999999999999998e-128

   1. Initial program 80.3%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.5%

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

   7. Simplified 75.5%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 84.1%

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

   9. Applied egg-rr 84.1%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. clear-num N/A

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. sub0-neg N/A

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

       9. distribute-frac-neg N/A

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

       10. remove-double-neg N/A

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

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

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

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

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

       13. associate-/l* N/A

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

       14. clear-num N/A

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

       15. un-div-inv N/A

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

       16. /-lowering-/.f64 N/A

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

       17. /-lowering-/.f64 92.8%

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

   11. Applied egg-rr 92.8%

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

   if 1.24999999999999992e145 < t1

   1. Initial program 45.4%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t1 around inf

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

   8. Simplified 85.9%

      \[\leadsto \frac{\color{blue}{v}}{-\left(u + t1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{u \cdot 2}{\frac{t1}{v}} - v}{t1}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-149}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(0 - t1\right) - u\right)}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{+145}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(0 - t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(0 - t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -9.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{t1 \cdot t\_1}{0 - t1}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;\left(0 - t1\right) \cdot \frac{t\_1}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= t1 -9.6e+96)
     (/ (* t1 t_1) (- 0.0 t1))
     (if (<= t1 1.6e+188) (* (- 0.0 t1) (/ t_1 (+ t1 u))) (/ v (- 0.0 t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (t1 <= -9.6e+96) {
		tmp = (t1 * t_1) / (0.0 - t1);
	} else if (t1 <= 1.6e+188) {
		tmp = (0.0 - t1) * (t_1 / (t1 + u));
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (t1 <= (-9.6d+96)) then
        tmp = (t1 * t_1) / (0.0d0 - t1)
    else if (t1 <= 1.6d+188) then
        tmp = (0.0d0 - t1) * (t_1 / (t1 + u))
    else
        tmp = v / (0.0d0 - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (t1 <= -9.6e+96) {
		tmp = (t1 * t_1) / (0.0 - t1);
	} else if (t1 <= 1.6e+188) {
		tmp = (0.0 - t1) * (t_1 / (t1 + u));
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if t1 <= -9.6e+96:
		tmp = (t1 * t_1) / (0.0 - t1)
	elif t1 <= 1.6e+188:
		tmp = (0.0 - t1) * (t_1 / (t1 + u))
	else:
		tmp = v / (0.0 - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -9.6e+96)
		tmp = Float64(Float64(t1 * t_1) / Float64(0.0 - t1));
	elseif (t1 <= 1.6e+188)
		tmp = Float64(Float64(0.0 - t1) * Float64(t_1 / Float64(t1 + u)));
	else
		tmp = Float64(v / Float64(0.0 - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -9.6e+96)
		tmp = (t1 * t_1) / (0.0 - t1);
	elseif (t1 <= 1.6e+188)
		tmp = (0.0 - t1) * (t_1 / (t1 + u));
	else
		tmp = v / (0.0 - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -9.6e+96], N[(N[(t1 * t$95$1), $MachinePrecision] / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.6e+188], N[(N[(0.0 - t1), $MachinePrecision] * N[(t$95$1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.59999999999999972e96

   1. Initial program 48.7%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 100.0%

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

   5. Simplified 100.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

      7. +-lowering-+.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in u around 0

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

   10. Simplified 92.0%

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

   if -9.59999999999999972e96 < t1 < 1.59999999999999985e188

   1. Initial program 80.4%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 92.6%

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

   4. Applied egg-rr 92.6%

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

   if 1.59999999999999985e188 < t1

   1. Initial program 45.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 92.2%

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

   5. Simplified 92.2%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 92.2%

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

   7. Applied egg-rr 92.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{0 - t1}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;\left(0 - t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-30}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(0 - t1\right) - u}}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+109}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{t1 + u}}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3e-30)
   (/ (* t1 (/ v (- (- 0.0 t1) u))) u)
   (if (<= u 9e+109)
     (/ (* v (/ t1 (+ t1 u))) (- 0.0 t1))
     (/ (/ t1 (- 0.0 u)) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3e-30) {
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u;
	} else if (u <= 9e+109) {
		tmp = (v * (t1 / (t1 + u))) / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3d-30)) then
        tmp = (t1 * (v / ((0.0d0 - t1) - u))) / u
    else if (u <= 9d+109) then
        tmp = (v * (t1 / (t1 + u))) / (0.0d0 - t1)
    else
        tmp = (t1 / (0.0d0 - u)) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3e-30) {
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u;
	} else if (u <= 9e+109) {
		tmp = (v * (t1 / (t1 + u))) / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3e-30:
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u
	elif u <= 9e+109:
		tmp = (v * (t1 / (t1 + u))) / (0.0 - t1)
	else:
		tmp = (t1 / (0.0 - u)) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3e-30)
		tmp = Float64(Float64(t1 * Float64(v / Float64(Float64(0.0 - t1) - u))) / u);
	elseif (u <= 9e+109)
		tmp = Float64(Float64(v * Float64(t1 / Float64(t1 + u))) / Float64(0.0 - t1));
	else
		tmp = Float64(Float64(t1 / Float64(0.0 - u)) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3e-30)
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u;
	elseif (u <= 9e+109)
		tmp = (v * (t1 / (t1 + u))) / (0.0 - t1);
	else
		tmp = (t1 / (0.0 - u)) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3e-30], N[(N[(t1 * N[(v / N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 9e+109], N[(N[(v * N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(0.0 - u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.9999999999999999e-30

   1. Initial program 75.0%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 95.4%

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

   5. Simplified 95.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in u around inf

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

   10. Simplified 89.3%

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

   if -2.9999999999999999e-30 < u < 8.9999999999999992e109

   1. Initial program 68.8%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 82.4%

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

   if 8.9999999999999992e109 < u

   1. Initial program 71.9%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 68.7%

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

   7. Simplified 68.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. associate-/l/ N/A

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

       6. times-frac N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

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

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

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

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

       12. /-lowering-/.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-30}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(0 - t1\right) - u}}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+109}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{t1 + u}}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(0 - t1\right) - u}}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 + u}{t1}}}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e-30)
   (/ (* t1 (/ v (- (- 0.0 t1) u))) u)
   (if (<= u 3.3e+109)
     (/ (/ v (/ (+ t1 u) t1)) (- 0.0 t1))
     (/ (/ t1 (- 0.0 u)) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-30) {
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u;
	} else if (u <= 3.3e+109) {
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.6d-30)) then
        tmp = (t1 * (v / ((0.0d0 - t1) - u))) / u
    else if (u <= 3.3d+109) then
        tmp = (v / ((t1 + u) / t1)) / (0.0d0 - t1)
    else
        tmp = (t1 / (0.0d0 - u)) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-30) {
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u;
	} else if (u <= 3.3e+109) {
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e-30:
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u
	elif u <= 3.3e+109:
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1)
	else:
		tmp = (t1 / (0.0 - u)) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e-30)
		tmp = Float64(Float64(t1 * Float64(v / Float64(Float64(0.0 - t1) - u))) / u);
	elseif (u <= 3.3e+109)
		tmp = Float64(Float64(v / Float64(Float64(t1 + u) / t1)) / Float64(0.0 - t1));
	else
		tmp = Float64(Float64(t1 / Float64(0.0 - u)) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e-30)
		tmp = (t1 * (v / ((0.0 - t1) - u))) / u;
	elseif (u <= 3.3e+109)
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1);
	else
		tmp = (t1 / (0.0 - u)) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e-30], N[(N[(t1 * N[(v / N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 3.3e+109], N[(N[(v / N[(N[(t1 + u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision] / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(0.0 - u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.59999999999999987e-30

   1. Initial program 75.0%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 95.4%

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

   5. Simplified 95.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in u around inf

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

   10. Simplified 89.3%

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

   if -2.59999999999999987e-30 < u < 3.2999999999999999e109

   1. Initial program 68.8%

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

   3. Taylor expanded in t1 around inf

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

   5. Simplified 54.4%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-frac-neg N/A

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

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

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

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

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. +-lowering-+.f64 82.4%

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

   7. Applied egg-rr 82.4%

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

   if 3.2999999999999999e109 < u

   1. Initial program 71.9%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 68.7%

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

   7. Simplified 68.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. associate-/l/ N/A

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

       6. times-frac N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

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

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

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

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

       12. /-lowering-/.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(0 - t1\right) - u}}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 + u}{t1}}}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;u \leq 6.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 + u}{t1}}}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e-29)
   (/ -1.0 (/ u (/ t1 (/ u v))))
   (if (<= u 6.8e+109)
     (/ (/ v (/ (+ t1 u) t1)) (- 0.0 t1))
     (/ (/ t1 (- 0.0 u)) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-29) {
		tmp = -1.0 / (u / (t1 / (u / v)));
	} else if (u <= 6.8e+109) {
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5d-29)) then
        tmp = (-1.0d0) / (u / (t1 / (u / v)))
    else if (u <= 6.8d+109) then
        tmp = (v / ((t1 + u) / t1)) / (0.0d0 - t1)
    else
        tmp = (t1 / (0.0d0 - u)) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-29) {
		tmp = -1.0 / (u / (t1 / (u / v)));
	} else if (u <= 6.8e+109) {
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e-29:
		tmp = -1.0 / (u / (t1 / (u / v)))
	elif u <= 6.8e+109:
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1)
	else:
		tmp = (t1 / (0.0 - u)) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e-29)
		tmp = Float64(-1.0 / Float64(u / Float64(t1 / Float64(u / v))));
	elseif (u <= 6.8e+109)
		tmp = Float64(Float64(v / Float64(Float64(t1 + u) / t1)) / Float64(0.0 - t1));
	else
		tmp = Float64(Float64(t1 / Float64(0.0 - u)) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5e-29)
		tmp = -1.0 / (u / (t1 / (u / v)));
	elseif (u <= 6.8e+109)
		tmp = (v / ((t1 + u) / t1)) / (0.0 - t1);
	else
		tmp = (t1 / (0.0 - u)) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e-29], N[(-1.0 / N[(u / N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.8e+109], N[(N[(v / N[(N[(t1 + u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision] / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(0.0 - u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.99999999999999986e-29

   1. Initial program 75.0%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 89.2%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 72.5%

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

   7. Simplified 72.5%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. clear-num N/A

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. sub0-neg N/A

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

       9. distribute-frac-neg N/A

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

       10. remove-double-neg N/A

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

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

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

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

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

       13. associate-/l* N/A

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

       14. clear-num N/A

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

       15. un-div-inv N/A

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

       16. /-lowering-/.f64 N/A

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

       17. /-lowering-/.f64 87.5%

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

   11. Applied egg-rr 87.5%

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

   if -4.99999999999999986e-29 < u < 6.80000000000000013e109

   1. Initial program 68.8%

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

   3. Taylor expanded in t1 around inf

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

   5. Simplified 54.4%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-frac-neg N/A

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

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

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

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

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. +-lowering-+.f64 82.4%

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

   7. Applied egg-rr 82.4%

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

   if 6.80000000000000013e109 < u

   1. Initial program 71.9%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 68.7%

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

   7. Simplified 68.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. associate-/l/ N/A

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

       6. times-frac N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

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

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

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

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

       12. /-lowering-/.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;u \leq 6.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 + u}{t1}}}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.95 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;u \leq 14000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.95e-33)
   (/ -1.0 (/ u (/ t1 (/ u v))))
   (if (<= u 14000000.0) (/ v (- 0.0 t1)) (/ (/ t1 (- 0.0 u)) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.95e-33) {
		tmp = -1.0 / (u / (t1 / (u / v)));
	} else if (u <= 14000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.95d-33)) then
        tmp = (-1.0d0) / (u / (t1 / (u / v)))
    else if (u <= 14000000.0d0) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = (t1 / (0.0d0 - u)) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.95e-33) {
		tmp = -1.0 / (u / (t1 / (u / v)));
	} else if (u <= 14000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = (t1 / (0.0 - u)) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.95e-33:
		tmp = -1.0 / (u / (t1 / (u / v)))
	elif u <= 14000000.0:
		tmp = v / (0.0 - t1)
	else:
		tmp = (t1 / (0.0 - u)) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.95e-33)
		tmp = Float64(-1.0 / Float64(u / Float64(t1 / Float64(u / v))));
	elseif (u <= 14000000.0)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = Float64(Float64(t1 / Float64(0.0 - u)) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.95e-33)
		tmp = -1.0 / (u / (t1 / (u / v)));
	elseif (u <= 14000000.0)
		tmp = v / (0.0 - t1);
	else
		tmp = (t1 / (0.0 - u)) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.95e-33], N[(-1.0 / N[(u / N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 14000000.0], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(0.0 - u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.94999999999999987e-33

   1. Initial program 75.4%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 72.9%

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

   7. Simplified 72.9%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. clear-num N/A

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. sub0-neg N/A

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

       9. distribute-frac-neg N/A

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

       10. remove-double-neg N/A

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

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

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

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

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

       13. associate-/l* N/A

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

       14. clear-num N/A

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

       15. un-div-inv N/A

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

       16. /-lowering-/.f64 N/A

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

       17. /-lowering-/.f64 87.7%

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

   11. Applied egg-rr 87.7%

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

   if -1.94999999999999987e-33 < u < 1.4e7

   1. Initial program 67.2%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 85.6%

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

   5. Simplified 85.6%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

   if 1.4e7 < u

   1. Initial program 73.3%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 83.5%

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

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 65.8%

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

   7. Simplified 65.8%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 72.9%

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

   9. Applied egg-rr 72.9%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. associate-/l/ N/A

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

       6. times-frac N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

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

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

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

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

       12. /-lowering-/.f64 76.3%

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

   11. Applied egg-rr 76.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.95 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;u \leq 14000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{0 - u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{if}\;u \leq -2.2 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3800000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ -1.0 (/ u (/ t1 (/ u v))))))
   (if (<= u -2.2e-33) t_1 (if (<= u 3800000000.0) (/ v (- 0.0 t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 / (u / (t1 / (u / v)));
	double tmp;
	if (u <= -2.2e-33) {
		tmp = t_1;
	} else if (u <= 3800000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-1.0d0) / (u / (t1 / (u / v)))
    if (u <= (-2.2d-33)) then
        tmp = t_1
    else if (u <= 3800000000.0d0) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -1.0 / (u / (t1 / (u / v)));
	double tmp;
	if (u <= -2.2e-33) {
		tmp = t_1;
	} else if (u <= 3800000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 / (u / (t1 / (u / v)))
	tmp = 0
	if u <= -2.2e-33:
		tmp = t_1
	elif u <= 3800000000.0:
		tmp = v / (0.0 - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 / Float64(u / Float64(t1 / Float64(u / v))))
	tmp = 0.0
	if (u <= -2.2e-33)
		tmp = t_1;
	elseif (u <= 3800000000.0)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 / (u / (t1 / (u / v)));
	tmp = 0.0;
	if (u <= -2.2e-33)
		tmp = t_1;
	elseif (u <= 3800000000.0)
		tmp = v / (0.0 - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 / N[(u / N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.2e-33], t$95$1, If[LessEqual[u, 3800000000.0], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.20000000000000005e-33 or 3.8e9 < u

   1. Initial program 74.3%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 86.3%

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

   4. Applied egg-rr 86.3%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 69.2%

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

   7. Simplified 69.2%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. distribute-neg-frac2 N/A

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

       2. sub0-neg N/A

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

       3. associate-*r/ N/A

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

       4. associate-/l* N/A

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

       5. clear-num N/A

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. sub0-neg N/A

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

       9. distribute-frac-neg N/A

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

       10. remove-double-neg N/A

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

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

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

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

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

       13. associate-/l* N/A

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

       14. clear-num N/A

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

       15. un-div-inv N/A

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

       16. /-lowering-/.f64 N/A

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

       17. /-lowering-/.f64 81.4%

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

   11. Applied egg-rr 81.4%

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

   if -2.20000000000000005e-33 < u < 3.8e9

   1. Initial program 67.2%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 85.6%

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

   5. Simplified 85.6%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \mathbf{elif}\;u \leq 3800000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{\frac{t1}{\frac{u}{v}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.95 \cdot 10^{-33}:\\ \;\;\;\;\frac{0 - t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 3600000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{0 - u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.95e-33)
   (/ (- 0.0 t1) (/ u (/ v u)))
   (if (<= u 3600000000.0) (/ v (- 0.0 t1)) (* t1 (/ (/ v (- 0.0 u)) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.95e-33) {
		tmp = (0.0 - t1) / (u / (v / u));
	} else if (u <= 3600000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t1 * ((v / (0.0 - u)) / 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 <= (-1.95d-33)) then
        tmp = (0.0d0 - t1) / (u / (v / u))
    else if (u <= 3600000000.0d0) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = t1 * ((v / (0.0d0 - u)) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.95e-33) {
		tmp = (0.0 - t1) / (u / (v / u));
	} else if (u <= 3600000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t1 * ((v / (0.0 - u)) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.95e-33:
		tmp = (0.0 - t1) / (u / (v / u))
	elif u <= 3600000000.0:
		tmp = v / (0.0 - t1)
	else:
		tmp = t1 * ((v / (0.0 - u)) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.95e-33)
		tmp = Float64(Float64(0.0 - t1) / Float64(u / Float64(v / u)));
	elseif (u <= 3600000000.0)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(0.0 - u)) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.95e-33)
		tmp = (0.0 - t1) / (u / (v / u));
	elseif (u <= 3600000000.0)
		tmp = v / (0.0 - t1);
	else
		tmp = t1 * ((v / (0.0 - u)) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.95e-33], N[(N[(0.0 - t1), $MachinePrecision] / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3600000000.0], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(0.0 - u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.94999999999999987e-33

   1. Initial program 75.4%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in t1 around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 72.9%

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

   7. Simplified 72.9%

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

      1. distribute-rgt-neg-out N/A

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

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

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. clear-num N/A

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

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

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

      10. /-lowering-/.f64 83.3%

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

   9. Applied egg-rr 83.3%

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

   if -1.94999999999999987e-33 < u < 3.6e9

   1. Initial program 67.2%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{t1}\right)}\right) \]

      4. /-lowering-/.f64 85.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{t1}\right)\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{t1}\right)\right) \]

      3. /-lowering-/.f64 85.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, t1\right)\right) \]

   7. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{-\frac{v}{t1}} \]

   if 3.6e9 < u

   1. Initial program 73.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto t1 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \color{blue}{\left(\mathsf{neg}\left(\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\right)}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\left(\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\left(\frac{\frac{v}{t1 + u}}{t1 + u}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{t1 + u}\right), \left(t1 + u\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(t1 + u\right)\right), \left(t1 + u\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \left(t1 + u\right)\right)\right)\right) \]

      10. +-lowering-+.f64 83.5%

          \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{+.f64}\left(t1, u\right)\right)\right)\right) \]

   4. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{\frac{v}{t1 + u}}{t1 + u}\right)} \]

   5. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\color{blue}{\left(\frac{v}{{u}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, \left({u}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, \left(u \cdot u\right)\right)\right)\right) \]

      3. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(u, u\right)\right)\right)\right) \]

   7. Simplified 65.8%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{v}{u \cdot u}}\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\left(\frac{\frac{v}{u}}{u}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{u}\right), u\right)\right)\right) \]

      3. /-lowering-/.f64 72.9%

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, u\right), u\right)\right)\right) \]

   9. Applied egg-rr 72.9%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{u}}{u}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.95 \cdot 10^{-33}:\\ \;\;\;\;\frac{0 - t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 3600000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{0 - u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0 - t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{if}\;u \leq -2.2 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 340000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- 0.0 t1) (/ u (/ v u)))))
   (if (<= u -2.2e-33) t_1 (if (<= u 340000000.0) (/ v (- 0.0 t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (0.0 - t1) / (u / (v / u));
	double tmp;
	if (u <= -2.2e-33) {
		tmp = t_1;
	} else if (u <= 340000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.0d0 - t1) / (u / (v / u))
    if (u <= (-2.2d-33)) then
        tmp = t_1
    else if (u <= 340000000.0d0) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (0.0 - t1) / (u / (v / u));
	double tmp;
	if (u <= -2.2e-33) {
		tmp = t_1;
	} else if (u <= 340000000.0) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (0.0 - t1) / (u / (v / u))
	tmp = 0
	if u <= -2.2e-33:
		tmp = t_1
	elif u <= 340000000.0:
		tmp = v / (0.0 - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(0.0 - t1) / Float64(u / Float64(v / u)))
	tmp = 0.0
	if (u <= -2.2e-33)
		tmp = t_1;
	elseif (u <= 340000000.0)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (0.0 - t1) / (u / (v / u));
	tmp = 0.0;
	if (u <= -2.2e-33)
		tmp = t_1;
	elseif (u <= 340000000.0)
		tmp = v / (0.0 - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(0.0 - t1), $MachinePrecision] / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.2e-33], t$95$1, If[LessEqual[u, 340000000.0], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.20000000000000005e-33 or 3.4e8 < u

   1. Initial program 74.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto t1 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \color{blue}{\left(\mathsf{neg}\left(\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\right)}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\left(\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\left(\frac{\frac{v}{t1 + u}}{t1 + u}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{t1 + u}\right), \left(t1 + u\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(t1 + u\right)\right), \left(t1 + u\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \left(t1 + u\right)\right)\right)\right) \]

      10. +-lowering-+.f64 86.3%

          \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(t1, u\right)\right), \mathsf{+.f64}\left(t1, u\right)\right)\right)\right) \]

   4. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{\frac{v}{t1 + u}}{t1 + u}\right)} \]

   5. Taylor expanded in t1 around 0

      \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\color{blue}{\left(\frac{v}{{u}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, \left({u}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, \left(u \cdot u\right)\right)\right)\right) \]

      3. *-lowering-*.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(t1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(u, u\right)\right)\right)\right) \]

   7. Simplified 69.2%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{v}{u \cdot u}}\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{neg}\left(t1 \cdot \frac{v}{u \cdot u}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(t1 \cdot \frac{v}{u \cdot u}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(t1 \cdot \frac{1}{\frac{u \cdot u}{v}}\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{t1}{\frac{u \cdot u}{v}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(t1, \left(\frac{u \cdot u}{v}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(t1, \left(\frac{1}{\frac{v}{u \cdot u}}\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(t1, \left(\frac{1}{\frac{\frac{v}{u}}{u}}\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(t1, \left(\frac{u}{\frac{v}{u}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(t1, \mathsf{/.f64}\left(u, \left(\frac{v}{u}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 77.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(t1, \mathsf{/.f64}\left(u, \mathsf{/.f64}\left(v, u\right)\right)\right)\right) \]

   9. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{-\frac{t1}{\frac{u}{\frac{v}{u}}}} \]

   if -2.20000000000000005e-33 < u < 3.4e8

   1. Initial program 67.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{t1}\right)}\right) \]

      4. /-lowering-/.f64 85.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{t1}\right)\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{t1}\right)\right) \]

      3. /-lowering-/.f64 85.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, t1\right)\right) \]

   7. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{-\frac{v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{0 - t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 340000000:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 8.6 \cdot 10^{+220}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+200)
   (/ -1.0 (/ u v))
   (if (<= u 8.6e+220) (/ v (- 0.0 t1)) (/ v (- 0.0 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+200) {
		tmp = -1.0 / (u / v);
	} else if (u <= 8.6e+220) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = v / (0.0 - 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 <= (-1.4d+200)) then
        tmp = (-1.0d0) / (u / v)
    else if (u <= 8.6d+220) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = v / (0.0d0 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+200) {
		tmp = -1.0 / (u / v);
	} else if (u <= 8.6e+220) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = v / (0.0 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+200:
		tmp = -1.0 / (u / v)
	elif u <= 8.6e+220:
		tmp = v / (0.0 - t1)
	else:
		tmp = v / (0.0 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+200)
		tmp = Float64(-1.0 / Float64(u / v));
	elseif (u <= 8.6e+220)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = Float64(v / Float64(0.0 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+200)
		tmp = -1.0 / (u / v);
	elseif (u <= 8.6e+220)
		tmp = v / (0.0 - t1);
	else
		tmp = v / (0.0 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+200], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.6e+220], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(0.0 - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.39999999999999992e200

   1. Initial program 68.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \color{blue}{t1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 48.3%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]

   6. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{u}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{u}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{u}\right)}\right) \]

      4. /-lowering-/.f64 42.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{u}\right)\right) \]

   8. Simplified 42.5%

      \[\leadsto \color{blue}{0 - \frac{v}{u}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{u}\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{u}{v}}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\frac{u}{v}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{u}}{v}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{u}{v}\right)}\right) \]

      6. /-lowering-/.f64 44.4%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(u, \color{blue}{v}\right)\right) \]

   10. Applied egg-rr 44.4%

       \[\leadsto \color{blue}{\frac{-1}{\frac{u}{v}}} \]

   if -1.39999999999999992e200 < u < 8.5999999999999999e220

   1. Initial program 70.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{t1}\right)}\right) \]

      4. /-lowering-/.f64 67.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{t1}\right)\right) \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{t1}\right)\right) \]

      3. /-lowering-/.f64 67.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, t1\right)\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{-\frac{v}{t1}} \]

   if 8.5999999999999999e220 < u

   1. Initial program 80.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \color{blue}{t1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 54.8%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]

   6. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{u}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{u}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{u}\right)}\right) \]

      4. /-lowering-/.f64 35.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{u}\right)\right) \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{0 - \frac{v}{u}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{u}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{u}\right)\right) \]

      3. /-lowering-/.f64 35.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, u\right)\right) \]

   10. Applied egg-rr 35.3%

       \[\leadsto \color{blue}{-\frac{v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 8.6 \cdot 10^{+220}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{0 - u}\\ \mathbf{if}\;u \leq -3.8 \cdot 10^{+200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+221}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- 0.0 u))))
   (if (<= u -3.8e+200) t_1 (if (<= u 9e+221) (/ v (- 0.0 t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (0.0 - u);
	double tmp;
	if (u <= -3.8e+200) {
		tmp = t_1;
	} else if (u <= 9e+221) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (0.0d0 - u)
    if (u <= (-3.8d+200)) then
        tmp = t_1
    else if (u <= 9d+221) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (0.0 - u);
	double tmp;
	if (u <= -3.8e+200) {
		tmp = t_1;
	} else if (u <= 9e+221) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (0.0 - u)
	tmp = 0
	if u <= -3.8e+200:
		tmp = t_1
	elif u <= 9e+221:
		tmp = v / (0.0 - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(0.0 - u))
	tmp = 0.0
	if (u <= -3.8e+200)
		tmp = t_1;
	elseif (u <= 9e+221)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (0.0 - u);
	tmp = 0.0;
	if (u <= -3.8e+200)
		tmp = t_1;
	elseif (u <= 9e+221)
		tmp = v / (0.0 - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(0.0 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.8e+200], t$95$1, If[LessEqual[u, 9e+221], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.79999999999999982e200 or 9.0000000000000004e221 < u

   1. Initial program 73.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(t1\right), v\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(t1, u\right), \color{blue}{t1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 51.2%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]

   6. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{u}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{u}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{u}\right)}\right) \]

      4. /-lowering-/.f64 39.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{u}\right)\right) \]

   8. Simplified 39.3%

      \[\leadsto \color{blue}{0 - \frac{v}{u}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{u}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{u}\right)\right) \]

      3. /-lowering-/.f64 39.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, u\right)\right) \]

   10. Applied egg-rr 39.3%

       \[\leadsto \color{blue}{-\frac{v}{u}} \]

   if -3.79999999999999982e200 < u < 9.0000000000000004e221

   1. Initial program 70.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{t1}\right)}\right) \]

      4. /-lowering-/.f64 67.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{t1}\right)\right) \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{t1}\right)\right) \]

      3. /-lowering-/.f64 67.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, t1\right)\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{-\frac{v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+200}:\\ \;\;\;\;\frac{v}{0 - u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+221}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.0% 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 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{t1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{t1 \cdot v}{t1 + u}}{t1 + u}\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \frac{\frac{t1 \cdot v}{t1 + u}}{\color{blue}{\mathsf{neg}\left(\left(t1 + u\right)\right)}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{t1 \cdot v}{t1 + u}\right), \color{blue}{\left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{v \cdot t1}{t1 + u}\right), \left(\mathsf{neg}\left(\left(\color{blue}{t1} + u\right)\right)\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot \frac{t1}{t1 + u}\right), \left(\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{t1}{t1 + u}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(t1 + u\right)}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \left(t1 + u\right)\right)\right), \left(\mathsf{neg}\left(\left(t1 + \color{blue}{u}\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \left(u + t1\right)\right)\right), \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \mathsf{+.f64}\left(u, t1\right)\right)\right), \left(\mathsf{neg}\left(\left(t1 + u\right)\right)\right)\right) \]

   12. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \mathsf{+.f64}\left(u, t1\right)\right)\right), \mathsf{neg.f64}\left(\left(t1 + u\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \mathsf{+.f64}\left(u, t1\right)\right)\right), \mathsf{neg.f64}\left(\left(u + t1\right)\right)\right) \]

   14. +-lowering-+.f64 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(t1, \mathsf{+.f64}\left(u, t1\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(u, t1\right)\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{v \cdot \frac{t1}{u + t1}}{-\left(u + t1\right)}} \]

6. Taylor expanded in t1 around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{v}, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(u, t1\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 64.4%

   \[\leadsto \frac{\color{blue}{v}}{-\left(u + t1\right)} \]

9. Final simplification 64.4%

   \[\leadsto \frac{v}{\left(0 - t1\right) - u} \]
10. Add Preprocessing

Alternative 14: 54.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{v}{0 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- 0.0 t1)))

C

double code(double u, double v, double t1) {
	return v / (0.0 - 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 / (0.0d0 - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (0.0 - t1);
}

Python

def code(u, v, t1):
	return v / (0.0 - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(0.0 - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (0.0 - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing

3. Taylor expanded in t1 around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{v}{t1}\right)}\right) \]

   4. /-lowering-/.f64 57.8%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(v, \color{blue}{t1}\right)\right) \]

5. Simplified 57.8%

   \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{v}{t1}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{v}{t1}\right)\right) \]

   3. /-lowering-/.f64 57.8%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(v, t1\right)\right) \]

7. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{-\frac{v}{t1}} \]

8. Final simplification 57.8%

   \[\leadsto \frac{v}{0 - t1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))