Rosa's DopplerBench

Percentage Accurate: 72.2% → 98.1%

Time: 10.8s

Alternatives: 10

Speedup: 0.8×

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: 72.2% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t1 + u\right) \cdot \left(t1 + u\right)\\ t_2 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -3.35 \cdot 10^{-130}:\\ \;\;\;\;v \cdot \frac{-t1}{t\_1}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{-283}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+86}:\\ \;\;\;\;t1 \cdot \left(v \cdot \frac{-1}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (+ t1 u) (+ t1 u))) (t_2 (/ v (- (- t1) u))))
   (if (<= t1 -1e+154)
     t_2
     (if (<= t1 -3.35e-130)
       (* v (/ (- t1) t_1))
       (if (<= t1 1.6e-283)
         (/ (* v (/ t1 u)) (- u))
         (if (<= t1 1.65e+86) (* t1 (* v (/ -1.0 t_1))) t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 + u) * (t1 + u);
	double t_2 = v / (-t1 - u);
	double tmp;
	if (t1 <= -1e+154) {
		tmp = t_2;
	} else if (t1 <= -3.35e-130) {
		tmp = v * (-t1 / t_1);
	} else if (t1 <= 1.6e-283) {
		tmp = (v * (t1 / u)) / -u;
	} else if (t1 <= 1.65e+86) {
		tmp = t1 * (v * (-1.0 / t_1));
	} else {
		tmp = t_2;
	}
	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 = (t1 + u) * (t1 + u)
    t_2 = v / (-t1 - u)
    if (t1 <= (-1d+154)) then
        tmp = t_2
    else if (t1 <= (-3.35d-130)) then
        tmp = v * (-t1 / t_1)
    else if (t1 <= 1.6d-283) then
        tmp = (v * (t1 / u)) / -u
    else if (t1 <= 1.65d+86) then
        tmp = t1 * (v * ((-1.0d0) / t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 + u) * (t1 + u);
	double t_2 = v / (-t1 - u);
	double tmp;
	if (t1 <= -1e+154) {
		tmp = t_2;
	} else if (t1 <= -3.35e-130) {
		tmp = v * (-t1 / t_1);
	} else if (t1 <= 1.6e-283) {
		tmp = (v * (t1 / u)) / -u;
	} else if (t1 <= 1.65e+86) {
		tmp = t1 * (v * (-1.0 / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 + u) * (t1 + u)
	t_2 = v / (-t1 - u)
	tmp = 0
	if t1 <= -1e+154:
		tmp = t_2
	elif t1 <= -3.35e-130:
		tmp = v * (-t1 / t_1)
	elif t1 <= 1.6e-283:
		tmp = (v * (t1 / u)) / -u
	elif t1 <= 1.65e+86:
		tmp = t1 * (v * (-1.0 / t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 + u) * Float64(t1 + u))
	t_2 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -1e+154)
		tmp = t_2;
	elseif (t1 <= -3.35e-130)
		tmp = Float64(v * Float64(Float64(-t1) / t_1));
	elseif (t1 <= 1.6e-283)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	elseif (t1 <= 1.65e+86)
		tmp = Float64(t1 * Float64(v * Float64(-1.0 / t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 + u) * (t1 + u);
	t_2 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -1e+154)
		tmp = t_2;
	elseif (t1 <= -3.35e-130)
		tmp = v * (-t1 / t_1);
	elseif (t1 <= 1.6e-283)
		tmp = (v * (t1 / u)) / -u;
	elseif (t1 <= 1.65e+86)
		tmp = t1 * (v * (-1.0 / t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1e+154], t$95$2, If[LessEqual[t1, -3.35e-130], N[(v * N[((-t1) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.6e-283], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 1.65e+86], N[(t1 * N[(v * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.00000000000000004e154 or 1.65e86 < t1

   1. Initial program 43.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity 88.3

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

   9. Applied egg-rr 88.3%

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

   if -1.00000000000000004e154 < t1 < -3.34999999999999993e-130

   1. Initial program 77.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.7

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

   4. Applied egg-rr 93.7%

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

   if -3.34999999999999993e-130 < t1 < 1.60000000000000006e-283

   1. Initial program 81.0%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.5

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

   7. Applied egg-rr 86.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. remove-double-neg N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.9

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

   9. Applied egg-rr 90.9%

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

   if 1.60000000000000006e-283 < t1 < 1.65e86

   1. Initial program 91.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. remove-double-neg N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. associate-/r* N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 94.6

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

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{t1 \cdot \left(v \cdot \frac{-1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq -3.35 \cdot 10^{-130}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{-283}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+86}:\\ \;\;\;\;t1 \cdot \left(v \cdot \frac{-1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ t_2 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{-130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 1.3 \cdot 10^{-283}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+110}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))) (t_2 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.05e+154)
     t_1
     (if (<= t1 -1.1e-130)
       t_2
       (if (<= t1 1.3e-283)
         (/ (* v (/ t1 u)) (- u))
         (if (<= t1 5.7e+110) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.05e+154) {
		tmp = t_1;
	} else if (t1 <= -1.1e-130) {
		tmp = t_2;
	} else if (t1 <= 1.3e-283) {
		tmp = (v * (t1 / u)) / -u;
	} else if (t1 <= 5.7e+110) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = v / (-t1 - u)
    t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.05d+154)) then
        tmp = t_1
    else if (t1 <= (-1.1d-130)) then
        tmp = t_2
    else if (t1 <= 1.3d-283) then
        tmp = (v * (t1 / u)) / -u
    else if (t1 <= 5.7d+110) then
        tmp = t_2
    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 / (-t1 - u);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.05e+154) {
		tmp = t_1;
	} else if (t1 <= -1.1e-130) {
		tmp = t_2;
	} else if (t1 <= 1.3e-283) {
		tmp = (v * (t1 / u)) / -u;
	} else if (t1 <= 5.7e+110) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.05e+154:
		tmp = t_1
	elif t1 <= -1.1e-130:
		tmp = t_2
	elif t1 <= 1.3e-283:
		tmp = (v * (t1 / u)) / -u
	elif t1 <= 5.7e+110:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	t_2 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -1.05e+154)
		tmp = t_1;
	elseif (t1 <= -1.1e-130)
		tmp = t_2;
	elseif (t1 <= 1.3e-283)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	elseif (t1 <= 5.7e+110)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -1.05e+154)
		tmp = t_1;
	elseif (t1 <= -1.1e-130)
		tmp = t_2;
	elseif (t1 <= 1.3e-283)
		tmp = (v * (t1 / u)) / -u;
	elseif (t1 <= 5.7e+110)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.05e+154], t$95$1, If[LessEqual[t1, -1.1e-130], t$95$2, If[LessEqual[t1, 1.3e-283], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 5.7e+110], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.04999999999999997e154 or 5.7000000000000002e110 < t1

   1. Initial program 42.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 89.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity 89.2

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

   9. Applied egg-rr 89.2%

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

   if -1.04999999999999997e154 < t1 < -1.0999999999999999e-130 or 1.3000000000000001e-283 < t1 < 5.7000000000000002e110

   1. Initial program 84.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.5

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

   4. Applied egg-rr 93.5%

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

   if -1.0999999999999999e-130 < t1 < 1.3000000000000001e-283

   1. Initial program 81.0%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.5

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

   7. Applied egg-rr 86.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. remove-double-neg N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.9

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

   9. Applied egg-rr 90.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{-130}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.3 \cdot 10^{-283}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+110}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ t_2 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{-130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-299}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+112}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))) (t_2 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -9.5e+153)
     t_1
     (if (<= t1 -1.1e-130)
       t_2
       (if (<= t1 1.9e-299)
         (* (- (/ t1 u)) (/ v u))
         (if (<= t1 1.4e+112) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -9.5e+153) {
		tmp = t_1;
	} else if (t1 <= -1.1e-130) {
		tmp = t_2;
	} else if (t1 <= 1.9e-299) {
		tmp = -(t1 / u) * (v / u);
	} else if (t1 <= 1.4e+112) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = v / (-t1 - u)
    t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-9.5d+153)) then
        tmp = t_1
    else if (t1 <= (-1.1d-130)) then
        tmp = t_2
    else if (t1 <= 1.9d-299) then
        tmp = -(t1 / u) * (v / u)
    else if (t1 <= 1.4d+112) then
        tmp = t_2
    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 / (-t1 - u);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -9.5e+153) {
		tmp = t_1;
	} else if (t1 <= -1.1e-130) {
		tmp = t_2;
	} else if (t1 <= 1.9e-299) {
		tmp = -(t1 / u) * (v / u);
	} else if (t1 <= 1.4e+112) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -9.5e+153:
		tmp = t_1
	elif t1 <= -1.1e-130:
		tmp = t_2
	elif t1 <= 1.9e-299:
		tmp = -(t1 / u) * (v / u)
	elif t1 <= 1.4e+112:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	t_2 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -9.5e+153)
		tmp = t_1;
	elseif (t1 <= -1.1e-130)
		tmp = t_2;
	elseif (t1 <= 1.9e-299)
		tmp = Float64(Float64(-Float64(t1 / u)) * Float64(v / u));
	elseif (t1 <= 1.4e+112)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -9.5e+153)
		tmp = t_1;
	elseif (t1 <= -1.1e-130)
		tmp = t_2;
	elseif (t1 <= 1.9e-299)
		tmp = -(t1 / u) * (v / u);
	elseif (t1 <= 1.4e+112)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -9.5e+153], t$95$1, If[LessEqual[t1, -1.1e-130], t$95$2, If[LessEqual[t1, 1.9e-299], N[((-N[(t1 / u), $MachinePrecision]) * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.4e+112], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.4999999999999995e153 or 1.4000000000000001e112 < t1

   1. Initial program 42.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 89.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity 89.2

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

   9. Applied egg-rr 89.2%

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

   if -9.4999999999999995e153 < t1 < -1.0999999999999999e-130 or 1.9000000000000001e-299 < t1 < 1.4000000000000001e112

   1. Initial program 84.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   if -1.0999999999999999e-130 < t1 < 1.9000000000000001e-299

   1. Initial program 80.2%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.2

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

   5. Simplified 80.2%

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

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

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

      2. *-commutative N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 90.1

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

   7. Applied egg-rr 90.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{-130}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-299}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+112}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+112}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -4.5e+153)
     t_1
     (if (<= t1 1.4e+112) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -4.5e+153) {
		tmp = t_1;
	} else if (t1 <= 1.4e+112) {
		tmp = v * (-t1 / ((t1 + 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 = v / (-t1 - u)
    if (t1 <= (-4.5d+153)) then
        tmp = t_1
    else if (t1 <= 1.4d+112) then
        tmp = v * (-t1 / ((t1 + 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 = v / (-t1 - u);
	double tmp;
	if (t1 <= -4.5e+153) {
		tmp = t_1;
	} else if (t1 <= 1.4e+112) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	tmp = 0
	if t1 <= -4.5e+153:
		tmp = t_1
	elif t1 <= 1.4e+112:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -4.5e+153)
		tmp = t_1;
	elseif (t1 <= 1.4e+112)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -4.5e+153)
		tmp = t_1;
	elseif (t1 <= 1.4e+112)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.5e+153], t$95$1, If[LessEqual[t1, 1.4e+112], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.5000000000000001e153 or 1.4000000000000001e112 < t1

   1. Initial program 42.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 89.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity 89.2

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

   9. Applied egg-rr 89.2%

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

   if -4.5000000000000001e153 < t1 < 1.4000000000000001e112

   1. Initial program 83.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 88.2

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

   4. Applied egg-rr 88.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+112}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{if}\;u \leq -0.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 34000000000:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ (/ v u) u))))
   (if (<= u -0.7) t_1 (if (<= u 34000000000.0) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * ((v / u) / u);
	double tmp;
	if (u <= -0.7) {
		tmp = t_1;
	} else if (u <= 34000000000.0) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 * ((v / u) / u)
    if (u <= (-0.7d0)) then
        tmp = t_1
    else if (u <= 34000000000.0d0) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 * ((v / u) / u);
	double tmp;
	if (u <= -0.7) {
		tmp = t_1;
	} else if (u <= 34000000000.0) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 * ((v / u) / u)
	tmp = 0
	if u <= -0.7:
		tmp = t_1
	elif u <= 34000000000.0:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(Float64(v / u) / u))
	tmp = 0.0
	if (u <= -0.7)
		tmp = t_1;
	elseif (u <= 34000000000.0)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 * ((v / u) / u);
	tmp = 0.0;
	if (u <= -0.7)
		tmp = t_1;
	elseif (u <= 34000000000.0)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -0.7], t$95$1, If[LessEqual[u, 34000000000.0], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -0.69999999999999996 or 3.4e10 < u

   1. Initial program 77.5%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.9

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

   5. Simplified 68.9%

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

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

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

      2. lift-*.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 72.4

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

   7. Applied egg-rr 72.4%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-neg.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 80.0

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

   9. Applied egg-rr 80.0%

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

   if -0.69999999999999996 < u < 3.4e10

   1. Initial program 65.0%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.7:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 34000000000:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -1.95 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -1.95e-84)
     t_1
     (if (<= t1 1.45e+54) (* t1 (/ v (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -1.95e-84) {
		tmp = t_1;
	} else if (t1 <= 1.45e+54) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (-t1 - u)
    if (t1 <= (-1.95d-84)) then
        tmp = t_1
    else if (t1 <= 1.45d+54) then
        tmp = t1 * (v / (u * -u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -1.95e-84) {
		tmp = t_1;
	} else if (t1 <= 1.45e+54) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	tmp = 0
	if t1 <= -1.95e-84:
		tmp = t_1
	elif t1 <= 1.45e+54:
		tmp = t1 * (v / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -1.95e-84)
		tmp = t_1;
	elseif (t1 <= 1.45e+54)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -1.95e-84)
		tmp = t_1;
	elseif (t1 <= 1.45e+54)
		tmp = t1 * (v / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.95e-84], t$95$1, If[LessEqual[t1, 1.45e+54], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.95000000000000011e-84 or 1.4499999999999999e54 < t1

   1. Initial program 56.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 79.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity 79.1

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

   9. Applied egg-rr 79.1%

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

   if -1.95000000000000011e-84 < t1 < 1.4499999999999999e54

   1. Initial program 87.2%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 79.4

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

   5. Simplified 79.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -1.4 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.35 \cdot 10^{+221}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v u))))
   (if (<= u -1.4e+138) t_1 (if (<= u 3.35e+221) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -1.4e+138) {
		tmp = t_1;
	} else if (u <= 3.35e+221) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(v / u)
    if (u <= (-1.4d+138)) then
        tmp = t_1
    else if (u <= 3.35d+221) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -1.4e+138) {
		tmp = t_1;
	} else if (u <= 3.35e+221) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / u)
	tmp = 0
	if u <= -1.4e+138:
		tmp = t_1
	elif u <= 3.35e+221:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / u))
	tmp = 0.0
	if (u <= -1.4e+138)
		tmp = t_1;
	elseif (u <= 3.35e+221)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / u);
	tmp = 0.0;
	if (u <= -1.4e+138)
		tmp = t_1;
	elseif (u <= 3.35e+221)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / u), $MachinePrecision])}, If[LessEqual[u, -1.4e+138], t$95$1, If[LessEqual[u, 3.35e+221], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4e138 or 3.35000000000000008e221 < 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. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 39.3%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 34.7

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

   10. Simplified 34.7%

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

   if -1.4e138 < u < 3.35000000000000008e221

   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 inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 59.4

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

   5. Simplified 59.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+138}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.35 \cdot 10^{+221}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return 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 = v / (-t1 - u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 56.8%

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

   1. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-rgt-identity 56.8

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

9. Applied egg-rr 56.8%

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

10. Final simplification 56.8%

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

Alternative 10: 17.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return -(v / 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 / u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 56.8%

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

8. Taylor expanded in t1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 15.0

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

10. Simplified 15.0%

    \[\leadsto \color{blue}{-\frac{v}{u}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))