Rosa's DopplerBench

Percentage Accurate: 72.9% → 97.6%

Time: 11.3s

Alternatives: 12

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: 97.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 97.7%

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

4. Applied egg-rr 97.7%

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

   1. sub0-neg N/A

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

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

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

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

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

   4. +-lowering-+.f64 97.7%

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

6. Applied egg-rr 97.7%

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

7. Final simplification 97.7%

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

Alternative 2: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)\\ \mathbf{if}\;u \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ -1.0 u) (* t1 (/ v (+ t1 u))))))
   (if (<= u -3.6e-11) t_1 (if (<= u 4.3e-7) (/ v (- 0.0 t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-1.0 / u) * (t1 * (v / (t1 + u)));
	double tmp;
	if (u <= -3.6e-11) {
		tmp = t_1;
	} else if (u <= 4.3e-7) {
		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 * (v / (t1 + u)))
    if (u <= (-3.6d-11)) then
        tmp = t_1
    else if (u <= 4.3d-7) 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 * (v / (t1 + u)));
	double tmp;
	if (u <= -3.6e-11) {
		tmp = t_1;
	} else if (u <= 4.3e-7) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-1.0 / u) * (t1 * (v / (t1 + u)))
	tmp = 0
	if u <= -3.6e-11:
		tmp = t_1
	elif u <= 4.3e-7:
		tmp = v / (0.0 - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-1.0 / u) * Float64(t1 * Float64(v / Float64(t1 + u))))
	tmp = 0.0
	if (u <= -3.6e-11)
		tmp = t_1;
	elseif (u <= 4.3e-7)
		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 * (v / (t1 + u)));
	tmp = 0.0;
	if (u <= -3.6e-11)
		tmp = t_1;
	elseif (u <= 4.3e-7)
		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[(-1.0 / u), $MachinePrecision] * N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.6e-11], t$95$1, If[LessEqual[u, 4.3e-7], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.59999999999999985e-11 or 4.3000000000000001e-7 < u

   1. Initial program 71.5%

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

      1. remove-double-neg N/A

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

      2. neg-mul-1 N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. associate-/l* N/A

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

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

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

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

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

      12. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 82.3%

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

   if -3.59999999999999985e-11 < u < 4.3000000000000001e-7

   1. Initial program 54.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 79.6%

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

   5. Simplified 79.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 79.6%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{t1 + u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.26 \cdot 10^{-11}:\\ \;\;\;\;0 - \frac{\frac{t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.26e-11)
   (- 0.0 (/ (/ t1 (/ u v)) u))
   (if (<= u 5e-8) (/ v (- 0.0 t1)) (* (/ -1.0 u) (* t1 (/ v u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.26e-11) {
		tmp = 0.0 - ((t1 / (u / v)) / u);
	} else if (u <= 5e-8) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = (-1.0 / u) * (t1 * (v / u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.26d-11)) then
        tmp = 0.0d0 - ((t1 / (u / v)) / u)
    else if (u <= 5d-8) then
        tmp = v / (0.0d0 - t1)
    else
        tmp = ((-1.0d0) / u) * (t1 * (v / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.26e-11) {
		tmp = 0.0 - ((t1 / (u / v)) / u);
	} else if (u <= 5e-8) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = (-1.0 / u) * (t1 * (v / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.26e-11:
		tmp = 0.0 - ((t1 / (u / v)) / u)
	elif u <= 5e-8:
		tmp = v / (0.0 - t1)
	else:
		tmp = (-1.0 / u) * (t1 * (v / u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.26e-11)
		tmp = Float64(0.0 - Float64(Float64(t1 / Float64(u / v)) / u));
	elseif (u <= 5e-8)
		tmp = Float64(v / Float64(0.0 - t1));
	else
		tmp = Float64(Float64(-1.0 / u) * Float64(t1 * Float64(v / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.26e-11)
		tmp = 0.0 - ((t1 / (u / v)) / u);
	elseif (u <= 5e-8)
		tmp = v / (0.0 - t1);
	else
		tmp = (-1.0 / u) * (t1 * (v / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.26e-11], N[(0.0 - N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5e-8], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / u), $MachinePrecision] * N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.26e-11

   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 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 77.9%

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

   5. Simplified 77.9%

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

      1. sub0-neg N/A

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

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

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

      3. *-lowering-*.f64 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. distribute-frac-neg N/A

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

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

      7. /-lowering-/.f64 81.9%

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

   9. Applied egg-rr 81.9%

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

   if -1.26e-11 < u < 4.9999999999999998e-8

   1. Initial program 54.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 79.6%

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

   5. Simplified 79.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 79.6%

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

   7. Applied egg-rr 79.6%

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

   if 4.9999999999999998e-8 < u

   1. Initial program 65.1%

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

      1. remove-double-neg N/A

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

      2. neg-mul-1 N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. associate-/l* N/A

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

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

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

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

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

      12. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 81.4%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 81.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.26 \cdot 10^{-11}:\\ \;\;\;\;0 - \frac{\frac{t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{\frac{t1}{\frac{u}{v}}}{u}\\ \mathbf{if}\;u \leq -2.4 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ (/ t1 (/ u v)) u))))
   (if (<= u -2.4e-10) t_1 (if (<= u 3.7e-7) (/ 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 <= -2.4e-10) {
		tmp = t_1;
	} else if (u <= 3.7e-7) {
		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 <= (-2.4d-10)) then
        tmp = t_1
    else if (u <= 3.7d-7) 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 <= -2.4e-10) {
		tmp = t_1;
	} else if (u <= 3.7e-7) {
		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 <= -2.4e-10:
		tmp = t_1
	elif u <= 3.7e-7:
		tmp = v / (0.0 - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(Float64(t1 / Float64(u / v)) / u))
	tmp = 0.0
	if (u <= -2.4e-10)
		tmp = t_1;
	elseif (u <= 3.7e-7)
		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 <= -2.4e-10)
		tmp = t_1;
	elseif (u <= 3.7e-7)
		tmp = v / (0.0 - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.4e-10], t$95$1, If[LessEqual[u, 3.7e-7], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.4e-10 or 3.70000000000000004e-7 < u

   1. Initial program 71.5%

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

   3. Taylor expanded in t1 around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

      1. sub0-neg N/A

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

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

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

      3. *-lowering-*.f64 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. distribute-frac-neg N/A

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

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

      7. /-lowering-/.f64 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -2.4e-10 < u < 3.70000000000000004e-7

   1. Initial program 54.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 79.6%

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

   5. Simplified 79.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 79.6%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{-10}:\\ \;\;\;\;0 - \frac{\frac{t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;u \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{\frac{t1}{\frac{u}{v}}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{v}{u}}{0 - \frac{u}{t1}}\\ \mathbf{if}\;u \leq -1.7 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (/ v u) (- 0.0 (/ u t1)))))
   (if (<= u -1.7e-12) t_1 (if (<= u 2.2e-9) (/ v (- 0.0 t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) / (0.0 - (u / t1));
	double tmp;
	if (u <= -1.7e-12) {
		tmp = t_1;
	} else if (u <= 2.2e-9) {
		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 / u) / (0.0d0 - (u / t1))
    if (u <= (-1.7d-12)) then
        tmp = t_1
    else if (u <= 2.2d-9) 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 / u) / (0.0 - (u / t1));
	double tmp;
	if (u <= -1.7e-12) {
		tmp = t_1;
	} else if (u <= 2.2e-9) {
		tmp = v / (0.0 - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) / (0.0 - (u / t1))
	tmp = 0
	if u <= -1.7e-12:
		tmp = t_1
	elif u <= 2.2e-9:
		tmp = v / (0.0 - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) / Float64(0.0 - Float64(u / t1)))
	tmp = 0.0
	if (u <= -1.7e-12)
		tmp = t_1;
	elseif (u <= 2.2e-9)
		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 / u) / (0.0 - (u / t1));
	tmp = 0.0;
	if (u <= -1.7e-12)
		tmp = t_1;
	elseif (u <= 2.2e-9)
		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[(v / u), $MachinePrecision] / N[(0.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.7e-12], t$95$1, If[LessEqual[u, 2.2e-9], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.7e-12 or 2.1999999999999998e-9 < u

   1. Initial program 71.5%

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

   3. Taylor expanded in t1 around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

      1. sub0-neg N/A

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

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

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

      3. *-lowering-*.f64 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-/l/ N/A

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

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

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

      3. sub0-neg N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. div-inv N/A

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

      8. sub0-neg N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 79.3%

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

   9. Applied egg-rr 79.3%

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

   if -1.7e-12 < u < 2.1999999999999998e-9

   1. Initial program 54.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 79.6%

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

   5. Simplified 79.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 79.6%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{v}{u}}{0 - \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{0 - \frac{u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{+85}:\\ \;\;\;\;0 - \frac{v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (+ t1 u)))))
   (if (<= t1 -5.2e-121)
     t_1
     (if (<= t1 4.2e+85) (- 0.0 (/ v (/ u (/ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -5.2e-121) {
		tmp = t_1;
	} else if (t1 <= 4.2e+85) {
		tmp = 0.0 - (v / (u / (t1 / u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0d0 - (v / (t1 + u))
    if (t1 <= (-5.2d-121)) then
        tmp = t_1
    else if (t1 <= 4.2d+85) then
        tmp = 0.0d0 - (v / (u / (t1 / u)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / (t1 + u));
	double tmp;
	if (t1 <= -5.2e-121) {
		tmp = t_1;
	} else if (t1 <= 4.2e+85) {
		tmp = 0.0 - (v / (u / (t1 / u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 0.0 - (v / (t1 + u))
	tmp = 0
	if t1 <= -5.2e-121:
		tmp = t_1
	elif t1 <= 4.2e+85:
		tmp = 0.0 - (v / (u / (t1 / u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -5.2e-121)
		tmp = t_1;
	elseif (t1 <= 4.2e+85)
		tmp = Float64(0.0 - Float64(v / Float64(u / Float64(t1 / u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 0.0 - (v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -5.2e-121)
		tmp = t_1;
	elseif (t1 <= 4.2e+85)
		tmp = 0.0 - (v / (u / (t1 / u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.2e-121], t$95$1, If[LessEqual[t1, 4.2e+85], N[(0.0 - N[(v / N[(u / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.19999999999999972e-121 or 4.2000000000000002e85 < t1

   1. Initial program 51.4%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. distribute-frac-neg N/A

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 79.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. +-lowering-+.f64 79.4%

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

   9. Applied egg-rr 79.4%

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

   if -5.19999999999999972e-121 < t1 < 4.2000000000000002e85

   1. Initial program 75.3%

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

   3. Taylor expanded in t1 around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. *-lowering-*.f64 69.4%

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

   5. Simplified 69.4%

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

      1. frac-2neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub0-neg N/A

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

      4. frac-2neg N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. neg-mul-1 N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

      12. /-lowering-/.f64 73.0%

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

   7. Applied egg-rr 73.0%

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

      1. frac-times N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. associate-*r/ N/A

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

      10. /-lowering-/.f64 73.0%

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

   9. Applied egg-rr 73.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-121}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{+85}:\\ \;\;\;\;0 - \frac{v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 97.7%

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

4. Applied egg-rr 97.7%

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

   1. sub0-neg N/A

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

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

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

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

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

   4. +-lowering-+.f64 97.7%

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

6. Applied egg-rr 97.7%

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

   1. frac-2neg N/A

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

   2. associate-/r/ N/A

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

   3. frac-2neg N/A

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

   4. neg-mul-1 N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f64 94.1%

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

8. Applied egg-rr 94.1%

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

   1. div-inv N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. remove-double-div N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. distribute-frac-neg N/A

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

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

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

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

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

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

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

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

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

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

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

   14. +-lowering-+.f64 94.0%

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

10. Applied egg-rr 94.0%

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

11. Final simplification 94.0%

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

Alternative 8: 56.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+164}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.4e+164) (/ -1.0 (/ u v)) (/ v (- 0.0 t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.4e+164) {
		tmp = -1.0 / (u / v);
	} 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) :: tmp
    if (u <= (-2.4d+164)) then
        tmp = (-1.0d0) / (u / v)
    else
        tmp = v / (0.0d0 - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.4e+164) {
		tmp = -1.0 / (u / v);
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.4e+164:
		tmp = -1.0 / (u / v)
	else:
		tmp = v / (0.0 - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.4e+164)
		tmp = Float64(-1.0 / Float64(u / v));
	else
		tmp = Float64(v / Float64(0.0 - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.4e+164)
		tmp = -1.0 / (u / v);
	else
		tmp = v / (0.0 - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.4e+164], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.40000000000000011e164

   1. Initial program 78.2%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. distribute-frac-neg N/A

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 47.2%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 40.2%

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

   if -2.40000000000000011e164 < u

   1. Initial program 60.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 59.5%

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

   5. Simplified 59.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 59.5%

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

   7. Applied egg-rr 59.5%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+164}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+166}:\\ \;\;\;\;v \cdot \frac{-1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.35e+166) (* v (/ -1.0 u)) (/ v (- 0.0 t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.35e+166) {
		tmp = v * (-1.0 / 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) :: tmp
    if (u <= (-1.35d+166)) then
        tmp = v * ((-1.0d0) / 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 tmp;
	if (u <= -1.35e+166) {
		tmp = v * (-1.0 / u);
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.35e+166:
		tmp = v * (-1.0 / u)
	else:
		tmp = v / (0.0 - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.35e+166)
		tmp = Float64(v * Float64(-1.0 / u));
	else
		tmp = Float64(v / Float64(0.0 - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.35e+166)
		tmp = v * (-1.0 / u);
	else
		tmp = v / (0.0 - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.35e+166], N[(v * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.35000000000000006e166

   1. Initial program 78.2%

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

      1. remove-double-neg N/A

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

      2. neg-mul-1 N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. associate-/l* N/A

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

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

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

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

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

      12. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 92.9%

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

   8. Taylor expanded in t1 around inf

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

   10. Simplified 38.1%

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

   if -1.35000000000000006e166 < u

   1. Initial program 60.5%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 59.5%

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

   5. Simplified 59.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 59.5%

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

   7. Applied egg-rr 59.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+166}:\\ \;\;\;\;v \cdot \frac{-1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-1.0d0) / ((t1 + u) / v)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 97.7%

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 61.1%

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

Alternative 11: 61.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 97.7%

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 61.1%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. mul-1-neg N/A

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

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

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

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

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

   6. +-lowering-+.f64 60.8%

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

9. Applied egg-rr 60.8%

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

10. Final simplification 60.8%

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

Alternative 12: 53.8% 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 62.4%

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

3. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. --lowering--.f64 54.8%

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

5. Simplified 54.8%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 54.8%

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

7. Applied egg-rr 54.8%

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

8. Final simplification 54.8%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))