Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.2%

Time: 11.2s

Alternatives: 11

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: 72.4% 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: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

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

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

   5. lift-neg.f64 N/A

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

   6. remove-double-neg N/A

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

   7. lower-*.f64 97.6

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

6. Applied rewrites 97.6%

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

7. Final simplification 97.6%

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

Alternative 2: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t1 + u\right) \cdot \left(t1 + u\right)\\ t_2 := -\frac{v}{t1}\\ \mathbf{if}\;t1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -5.3 \cdot 10^{-182}:\\ \;\;\;\;v \cdot \frac{-t1}{t\_1}\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{-246}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;t1 \cdot \left(v \cdot \frac{-1}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (+ t1 u) (+ t1 u))) (t_2 (- (/ v t1))))
   (if (<= t1 -5e+154)
     t_2
     (if (<= t1 -5.3e-182)
       (* v (/ (- t1) t_1))
       (if (<= t1 1.7e-246)
         (* v (/ (/ t1 (- u)) (+ t1 u)))
         (if (<= t1 1.95e+149) (* 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);
	double tmp;
	if (t1 <= -5e+154) {
		tmp = t_2;
	} else if (t1 <= -5.3e-182) {
		tmp = v * (-t1 / t_1);
	} else if (t1 <= 1.7e-246) {
		tmp = v * ((t1 / -u) / (t1 + u));
	} else if (t1 <= 1.95e+149) {
		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)
    if (t1 <= (-5d+154)) then
        tmp = t_2
    else if (t1 <= (-5.3d-182)) then
        tmp = v * (-t1 / t_1)
    else if (t1 <= 1.7d-246) then
        tmp = v * ((t1 / -u) / (t1 + u))
    else if (t1 <= 1.95d+149) 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);
	double tmp;
	if (t1 <= -5e+154) {
		tmp = t_2;
	} else if (t1 <= -5.3e-182) {
		tmp = v * (-t1 / t_1);
	} else if (t1 <= 1.7e-246) {
		tmp = v * ((t1 / -u) / (t1 + u));
	} else if (t1 <= 1.95e+149) {
		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)
	tmp = 0
	if t1 <= -5e+154:
		tmp = t_2
	elif t1 <= -5.3e-182:
		tmp = v * (-t1 / t_1)
	elif t1 <= 1.7e-246:
		tmp = v * ((t1 / -u) / (t1 + u))
	elif t1 <= 1.95e+149:
		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(-Float64(v / t1))
	tmp = 0.0
	if (t1 <= -5e+154)
		tmp = t_2;
	elseif (t1 <= -5.3e-182)
		tmp = Float64(v * Float64(Float64(-t1) / t_1));
	elseif (t1 <= 1.7e-246)
		tmp = Float64(v * Float64(Float64(t1 / Float64(-u)) / Float64(t1 + u)));
	elseif (t1 <= 1.95e+149)
		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);
	tmp = 0.0;
	if (t1 <= -5e+154)
		tmp = t_2;
	elseif (t1 <= -5.3e-182)
		tmp = v * (-t1 / t_1);
	elseif (t1 <= 1.7e-246)
		tmp = v * ((t1 / -u) / (t1 + u));
	elseif (t1 <= 1.95e+149)
		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 / t1), $MachinePrecision])}, If[LessEqual[t1, -5e+154], t$95$2, If[LessEqual[t1, -5.3e-182], N[(v * N[((-t1) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.7e-246], N[(v * N[(N[(t1 / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.95e+149], 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 < -5.00000000000000004e154 or 1.95e149 < t1

   1. Initial program 30.4%

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

   3. Taylor expanded in t1 around inf

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

      1. 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 93.3

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

   5. Applied rewrites 93.3%

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

   if -5.00000000000000004e154 < t1 < -5.30000000000000005e-182

   1. Initial program 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.4

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

   4. Applied rewrites 88.4%

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

   if -5.30000000000000005e-182 < t1 < 1.7000000000000001e-246

   1. Initial program 72.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 95.1

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

      12. lift-neg.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-neg-in N/A

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

      15. lift-neg.f64 N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 95.1

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.0

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

   9. Applied rewrites 91.0%

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

   if 1.7000000000000001e-246 < t1 < 1.95e149

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. remove-double-neg N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq -5.3 \cdot 10^{-182}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{-246}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;t1 \cdot \left(v \cdot \frac{-1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1}\\ t_2 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -2.4 \cdot 10^{-231}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v t1))) (t_2 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -5e+154)
     t_1
     (if (<= t1 -2.4e-231)
       t_2
       (if (<= t1 2.1e-125)
         (/ (* t1 (/ v u)) (- u))
         (if (<= t1 5.7e+149) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / t1);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -5e+154) {
		tmp = t_1;
	} else if (t1 <= -2.4e-231) {
		tmp = t_2;
	} else if (t1 <= 2.1e-125) {
		tmp = (t1 * (v / u)) / -u;
	} else if (t1 <= 5.7e+149) {
		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)
    t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-5d+154)) then
        tmp = t_1
    else if (t1 <= (-2.4d-231)) then
        tmp = t_2
    else if (t1 <= 2.1d-125) then
        tmp = (t1 * (v / u)) / -u
    else if (t1 <= 5.7d+149) 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);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -5e+154) {
		tmp = t_1;
	} else if (t1 <= -2.4e-231) {
		tmp = t_2;
	} else if (t1 <= 2.1e-125) {
		tmp = (t1 * (v / u)) / -u;
	} else if (t1 <= 5.7e+149) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / t1)
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -5e+154:
		tmp = t_1
	elif t1 <= -2.4e-231:
		tmp = t_2
	elif t1 <= 2.1e-125:
		tmp = (t1 * (v / u)) / -u
	elif t1 <= 5.7e+149:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / t1))
	t_2 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -5e+154)
		tmp = t_1;
	elseif (t1 <= -2.4e-231)
		tmp = t_2;
	elseif (t1 <= 2.1e-125)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	elseif (t1 <= 5.7e+149)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / t1);
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -5e+154)
		tmp = t_1;
	elseif (t1 <= -2.4e-231)
		tmp = t_2;
	elseif (t1 <= 2.1e-125)
		tmp = (t1 * (v / u)) / -u;
	elseif (t1 <= 5.7e+149)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / t1), $MachinePrecision])}, Block[{t$95$2 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5e+154], t$95$1, If[LessEqual[t1, -2.4e-231], t$95$2, If[LessEqual[t1, 2.1e-125], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 5.7e+149], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.00000000000000004e154 or 5.69999999999999965e149 < t1

   1. Initial program 30.4%

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

   3. Taylor expanded in t1 around inf

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

      1. 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 93.3

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

   5. Applied rewrites 93.3%

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

   if -5.00000000000000004e154 < t1 < -2.39999999999999992e-231 or 2.1e-125 < t1 < 5.69999999999999965e149

   1. Initial program 85.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if -2.39999999999999992e-231 < t1 < 2.1e-125

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in t1 around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.6

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

   10. Applied rewrites 85.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq -2.4 \cdot 10^{-231}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+149}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-247}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -2e+67)
     t_1
     (if (<= t1 4.8e-247)
       (* v (/ (- t1) (* u u)))
       (if (<= t1 9.5e-6) (* (- 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 <= -2e+67) {
		tmp = t_1;
	} else if (t1 <= 4.8e-247) {
		tmp = v * (-t1 / (u * u));
	} else if (t1 <= 9.5e-6) {
		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 <= (-2d+67)) then
        tmp = t_1
    else if (t1 <= 4.8d-247) then
        tmp = v * (-t1 / (u * u))
    else if (t1 <= 9.5d-6) 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 <= -2e+67) {
		tmp = t_1;
	} else if (t1 <= 4.8e-247) {
		tmp = v * (-t1 / (u * u));
	} else if (t1 <= 9.5e-6) {
		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 <= -2e+67:
		tmp = t_1
	elif t1 <= 4.8e-247:
		tmp = v * (-t1 / (u * u))
	elif t1 <= 9.5e-6:
		tmp = -t1 * (v / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -2e+67)
		tmp = t_1;
	elseif (t1 <= 4.8e-247)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	elseif (t1 <= 9.5e-6)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * 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 <= -2e+67)
		tmp = t_1;
	elseif (t1 <= 4.8e-247)
		tmp = v * (-t1 / (u * u));
	elseif (t1 <= 9.5e-6)
		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, -2e+67], t$95$1, If[LessEqual[t1, 4.8e-247], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 9.5e-6], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.99999999999999997e67 or 9.5000000000000005e-6 < t1

   1. Initial program 53.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 22.1

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

   7. Applied rewrites 22.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 22.1

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

   9. Applied rewrites 18.0%

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

   10. Taylor expanded in t1 around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 88.6

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

   12. Applied rewrites 88.6%

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

   if -1.99999999999999997e67 < t1 < 4.80000000000000022e-247

   1. Initial program 80.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if 4.80000000000000022e-247 < t1 < 9.5000000000000005e-6

   1. Initial program 84.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 73.7

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

   5. Applied rewrites 73.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-247}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1}\\ \mathbf{if}\;t1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+149}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v t1))))
   (if (<= t1 -5e+154)
     t_1
     (if (<= t1 5.7e+149) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / t1);
	double tmp;
	if (t1 <= -5e+154) {
		tmp = t_1;
	} else if (t1 <= 5.7e+149) {
		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)
    if (t1 <= (-5d+154)) then
        tmp = t_1
    else if (t1 <= 5.7d+149) 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);
	double tmp;
	if (t1 <= -5e+154) {
		tmp = t_1;
	} else if (t1 <= 5.7e+149) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / t1)
	tmp = 0
	if t1 <= -5e+154:
		tmp = t_1
	elif t1 <= 5.7e+149:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / t1))
	tmp = 0.0
	if (t1 <= -5e+154)
		tmp = t_1;
	elseif (t1 <= 5.7e+149)
		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);
	tmp = 0.0;
	if (t1 <= -5e+154)
		tmp = t_1;
	elseif (t1 <= 5.7e+149)
		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 / t1), $MachinePrecision])}, If[LessEqual[t1, -5e+154], t$95$1, If[LessEqual[t1, 5.7e+149], 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 < -5.00000000000000004e154 or 5.69999999999999965e149 < t1

   1. Initial program 30.4%

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

   3. Taylor expanded in t1 around inf

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

      1. 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 93.3

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

   5. Applied rewrites 93.3%

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

   if -5.00000000000000004e154 < t1 < 5.69999999999999965e149

   1. Initial program 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 84.8

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

   4. Applied rewrites 84.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+149}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -2e+67) t_1 (if (<= t1 9.5e-6) (/ (* 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 <= -2e+67) {
		tmp = t_1;
	} else if (t1 <= 9.5e-6) {
		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 <= (-2d+67)) then
        tmp = t_1
    else if (t1 <= 9.5d-6) 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 <= -2e+67) {
		tmp = t_1;
	} else if (t1 <= 9.5e-6) {
		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 <= -2e+67:
		tmp = t_1
	elif t1 <= 9.5e-6:
		tmp = (t1 * -v) / (u * u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -2e+67)
		tmp = t_1;
	elseif (t1 <= 9.5e-6)
		tmp = Float64(Float64(t1 * Float64(-v)) / Float64(u * 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 <= -2e+67)
		tmp = t_1;
	elseif (t1 <= 9.5e-6)
		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, -2e+67], t$95$1, If[LessEqual[t1, 9.5e-6], N[(N[(t1 * (-v)), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.99999999999999997e67 or 9.5000000000000005e-6 < t1

   1. Initial program 53.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 22.1

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

   7. Applied rewrites 22.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 22.1

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

   9. Applied rewrites 18.0%

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

   10. Taylor expanded in t1 around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 88.6

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

   12. Applied rewrites 88.6%

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

   if -1.99999999999999997e67 < t1 < 9.5000000000000005e-6

   1. Initial program 81.6%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -2e+67) t_1 (if (<= t1 9.5e-6) (* (- 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 <= -2e+67) {
		tmp = t_1;
	} else if (t1 <= 9.5e-6) {
		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 <= (-2d+67)) then
        tmp = t_1
    else if (t1 <= 9.5d-6) 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 <= -2e+67) {
		tmp = t_1;
	} else if (t1 <= 9.5e-6) {
		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 <= -2e+67:
		tmp = t_1
	elif t1 <= 9.5e-6:
		tmp = -t1 * (v / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -2e+67)
		tmp = t_1;
	elseif (t1 <= 9.5e-6)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * 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 <= -2e+67)
		tmp = t_1;
	elseif (t1 <= 9.5e-6)
		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, -2e+67], t$95$1, If[LessEqual[t1, 9.5e-6], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.99999999999999997e67 or 9.5000000000000005e-6 < t1

   1. Initial program 53.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 22.1

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

   7. Applied rewrites 22.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 22.1

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

   9. Applied rewrites 18.0%

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

   10. Taylor expanded in t1 around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 88.6

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

   12. Applied rewrites 88.6%

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

   if -1.99999999999999997e67 < t1 < 9.5000000000000005e-6

   1. Initial program 81.6%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   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 96.7

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

4. Applied rewrites 96.7%

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

5. Final simplification 96.7%

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

Alternative 9: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{t1}{u \cdot u}\\ \mathbf{if}\;u \leq -2.4 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+201}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ t1 (* u u)))))
   (if (<= u -2.4e+157) t_1 (if (<= u 9e+201) (/ (- v) (+ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (t1 / (u * u));
	double tmp;
	if (u <= -2.4e+157) {
		tmp = t_1;
	} else if (u <= 9e+201) {
		tmp = -v / (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 * u))
    if (u <= (-2.4d+157)) then
        tmp = t_1
    else if (u <= 9d+201) then
        tmp = -v / (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 * u));
	double tmp;
	if (u <= -2.4e+157) {
		tmp = t_1;
	} else if (u <= 9e+201) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (t1 / (u * u))
	tmp = 0
	if u <= -2.4e+157:
		tmp = t_1
	elif u <= 9e+201:
		tmp = -v / (t1 + u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(t1 / Float64(u * u)))
	tmp = 0.0
	if (u <= -2.4e+157)
		tmp = t_1;
	elseif (u <= 9e+201)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (t1 / (u * u));
	tmp = 0.0;
	if (u <= -2.4e+157)
		tmp = t_1;
	elseif (u <= 9e+201)
		tmp = -v / (t1 + u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.4e+157], t$95$1, If[LessEqual[u, 9e+201], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.4e157 or 9.0000000000000002e201 < u

   1. Initial program 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 72.2

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 72.2%

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

   if -2.4e157 < u < 9.0000000000000002e201

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in t1 around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 40.4

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

   7. Applied rewrites 40.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 40.4

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

   9. Applied rewrites 38.2%

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

   10. Taylor expanded in t1 around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 64.8

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

   12. Applied rewrites 64.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+157}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+201}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 1.8× speedup

Math

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

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(Float64(-v) / 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 68.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 97.6

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in t1 around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 51.3

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

7. Applied rewrites 51.3%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg2 N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 51.3

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

9. Applied rewrites 48.4%

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

10. Taylor expanded in t1 around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 58.9

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

12. Applied rewrites 58.9%

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

Alternative 11: 53.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 53.3

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

5. Applied rewrites 53.3%

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

6. Final simplification 53.3%

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

Reproduce

herbie shell --seed 2024232 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))