Rosa's DopplerBench

Percentage Accurate: 73.5% → 98.1%

Time: 10.8s

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% 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 69.3%

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

   1. lift-/.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 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\left(-\frac{t1}{t1 + u}\right) \cdot \left(-v\right)}{-\left(t1 + u\right)}} \]
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 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+159}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -4.7 \cdot 10^{-155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -2.6e+159)
     (- (* 1.0 (/ v (+ t1 u))))
     (if (<= t1 -4.7e-155)
       t_1
       (if (<= t1 2.5e-188)
         (/ (* v (/ t1 u)) (- (- t1) u))
         (if (<= t1 5.8e+75) t_1 (* (/ t1 (+ t1 u)) (/ v (- t1)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -2.6e+159) {
		tmp = -(1.0 * (v / (t1 + u)));
	} else if (t1 <= -4.7e-155) {
		tmp = t_1;
	} else if (t1 <= 2.5e-188) {
		tmp = (v * (t1 / u)) / (-t1 - u);
	} else if (t1 <= 5.8e+75) {
		tmp = t_1;
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-2.6d+159)) then
        tmp = -(1.0d0 * (v / (t1 + u)))
    else if (t1 <= (-4.7d-155)) then
        tmp = t_1
    else if (t1 <= 2.5d-188) then
        tmp = (v * (t1 / u)) / (-t1 - u)
    else if (t1 <= 5.8d+75) then
        tmp = t_1
    else
        tmp = (t1 / (t1 + u)) * (v / -t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -2.6e+159) {
		tmp = -(1.0 * (v / (t1 + u)));
	} else if (t1 <= -4.7e-155) {
		tmp = t_1;
	} else if (t1 <= 2.5e-188) {
		tmp = (v * (t1 / u)) / (-t1 - u);
	} else if (t1 <= 5.8e+75) {
		tmp = t_1;
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -2.6e+159:
		tmp = -(1.0 * (v / (t1 + u)))
	elif t1 <= -4.7e-155:
		tmp = t_1
	elif t1 <= 2.5e-188:
		tmp = (v * (t1 / u)) / (-t1 - u)
	elif t1 <= 5.8e+75:
		tmp = t_1
	else:
		tmp = (t1 / (t1 + u)) * (v / -t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -2.6e+159)
		tmp = Float64(-Float64(1.0 * Float64(v / Float64(t1 + u))));
	elseif (t1 <= -4.7e-155)
		tmp = t_1;
	elseif (t1 <= 2.5e-188)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(Float64(-t1) - u));
	elseif (t1 <= 5.8e+75)
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -2.6e+159)
		tmp = -(1.0 * (v / (t1 + u)));
	elseif (t1 <= -4.7e-155)
		tmp = t_1;
	elseif (t1 <= 2.5e-188)
		tmp = (v * (t1 / u)) / (-t1 - u);
	elseif (t1 <= 5.8e+75)
		tmp = t_1;
	else
		tmp = (t1 / (t1 + u)) * (v / -t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.6e+159], (-N[(1.0 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t1, -4.7e-155], t$95$1, If[LessEqual[t1, 2.5e-188], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.8e+75], t$95$1, N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.6e159

   1. Initial program 39.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   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. Applied rewrites 97.8%

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

   if -2.6e159 < t1 < -4.6999999999999998e-155 or 2.5e-188 < t1 < 5.7999999999999997e75

   1. Initial program 84.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-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 94.5

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

   4. Applied rewrites 94.5%

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

   if -4.6999999999999998e-155 < t1 < 2.5e-188

   1. Initial program 78.2%

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

      1. 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 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\left(-\frac{t1}{t1 + u}\right) \cdot \left(-v\right)}{-\left(t1 + u\right)}} \]
   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 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in t1 around 0

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

      1. lower-/.f64 94.5

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

   9. Applied rewrites 94.5%

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

   if 5.7999999999999997e75 < t1

   1. Initial program 47.5%

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

      1. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.4

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

   7. Applied rewrites 84.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+159}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -4.7 \cdot 10^{-155}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+159}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.6e+159)
   (- (* 1.0 (/ v (+ t1 u))))
   (if (<= t1 5.8e+75)
     (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))
     (* (/ t1 (+ t1 u)) (/ v (- t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.6e+159) {
		tmp = -(1.0 * (v / (t1 + u)));
	} else if (t1 <= 5.8e+75) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.6d+159)) then
        tmp = -(1.0d0 * (v / (t1 + u)))
    else if (t1 <= 5.8d+75) then
        tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
    else
        tmp = (t1 / (t1 + u)) * (v / -t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.6e+159) {
		tmp = -(1.0 * (v / (t1 + u)));
	} else if (t1 <= 5.8e+75) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.6e+159:
		tmp = -(1.0 * (v / (t1 + u)))
	elif t1 <= 5.8e+75:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = (t1 / (t1 + u)) * (v / -t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.6e+159)
		tmp = Float64(-Float64(1.0 * Float64(v / Float64(t1 + u))));
	elseif (t1 <= 5.8e+75)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.6e+159)
		tmp = -(1.0 * (v / (t1 + u)));
	elseif (t1 <= 5.8e+75)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = (t1 / (t1 + u)) * (v / -t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.6e+159], (-N[(1.0 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t1, 5.8e+75], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.6e159

   1. Initial program 39.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   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. Applied rewrites 97.8%

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

   if -2.6e159 < t1 < 5.7999999999999997e75

   1. Initial program 82.3%

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

      1. lift-/.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 88.5

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

   4. Applied rewrites 88.5%

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

   if 5.7999999999999997e75 < t1

   1. Initial program 47.5%

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

      1. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.4

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

   7. Applied rewrites 84.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+159}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (* 1.0 (/ v (+ t1 u))))))
   (if (<= t1 -2.6e+159)
     t_1
     (if (<= t1 5.8e+75) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -2.6e+159) {
		tmp = t_1;
	} else if (t1 <= 5.8e+75) {
		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 = -(1.0d0 * (v / (t1 + u)))
    if (t1 <= (-2.6d+159)) then
        tmp = t_1
    else if (t1 <= 5.8d+75) 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 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -2.6e+159) {
		tmp = t_1;
	} else if (t1 <= 5.8e+75) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(1.0 * (v / (t1 + u)))
	tmp = 0
	if t1 <= -2.6e+159:
		tmp = t_1
	elif t1 <= 5.8e+75:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(1.0 * Float64(v / Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -2.6e+159)
		tmp = t_1;
	elseif (t1 <= 5.8e+75)
		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 = -(1.0 * (v / (t1 + u)));
	tmp = 0.0;
	if (t1 <= -2.6e+159)
		tmp = t_1;
	elseif (t1 <= 5.8e+75)
		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[(1.0 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -2.6e+159], t$95$1, If[LessEqual[t1, 5.8e+75], 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 < -2.6e159 or 5.7999999999999997e75 < t1

   1. Initial program 44.5%

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

      1. 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 99.9

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

   4. Applied rewrites 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. Applied rewrites 89.6%

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

   if -2.6e159 < t1 < 5.7999999999999997e75

   1. Initial program 82.3%

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

      1. lift-/.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 88.5

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

   4. Applied rewrites 88.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+159}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-35}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(-\frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (* 1.0 (/ v (+ t1 u))))))
   (if (<= t1 -3.1e-145) t_1 (if (<= t1 9e-35) (* (/ t1 u) (- (/ v u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -3.1e-145) {
		tmp = t_1;
	} else if (t1 <= 9e-35) {
		tmp = (t1 / u) * -(v / 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 = -(1.0d0 * (v / (t1 + u)))
    if (t1 <= (-3.1d-145)) then
        tmp = t_1
    else if (t1 <= 9d-35) then
        tmp = (t1 / u) * -(v / 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 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -3.1e-145) {
		tmp = t_1;
	} else if (t1 <= 9e-35) {
		tmp = (t1 / u) * -(v / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(1.0 * (v / (t1 + u)))
	tmp = 0
	if t1 <= -3.1e-145:
		tmp = t_1
	elif t1 <= 9e-35:
		tmp = (t1 / u) * -(v / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(1.0 * Float64(v / Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3.1e-145)
		tmp = t_1;
	elseif (t1 <= 9e-35)
		tmp = Float64(Float64(t1 / u) * Float64(-Float64(v / u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(1.0 * (v / (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3.1e-145)
		tmp = t_1;
	elseif (t1 <= 9e-35)
		tmp = (t1 / u) * -(v / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(1.0 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -3.1e-145], t$95$1, If[LessEqual[t1, 9e-35], N[(N[(t1 / u), $MachinePrecision] * (-N[(v / u), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.1e-145 or 9.0000000000000002e-35 < t1

   1. Initial program 61.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 99.9

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

   4. Applied rewrites 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. Applied rewrites 80.1%

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

   if -3.1e-145 < t1 < 9.0000000000000002e-35

   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. 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. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.8

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in t1 around 0

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

      1. lower-/.f64 85.3

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

   10. Applied rewrites 85.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-35}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(-\frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (* 1.0 (/ v (+ t1 u))))))
   (if (<= t1 -3.1e-145)
     t_1
     (if (<= t1 8.5e-113) (* v (/ (- t1) (* u u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -3.1e-145) {
		tmp = t_1;
	} else if (t1 <= 8.5e-113) {
		tmp = v * (-t1 / (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 = -(1.0d0 * (v / (t1 + u)))
    if (t1 <= (-3.1d-145)) then
        tmp = t_1
    else if (t1 <= 8.5d-113) then
        tmp = v * (-t1 / (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 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -3.1e-145) {
		tmp = t_1;
	} else if (t1 <= 8.5e-113) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(1.0 * (v / (t1 + u)))
	tmp = 0
	if t1 <= -3.1e-145:
		tmp = t_1
	elif t1 <= 8.5e-113:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(1.0 * Float64(v / Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3.1e-145)
		tmp = t_1;
	elseif (t1 <= 8.5e-113)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(1.0 * (v / (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3.1e-145)
		tmp = t_1;
	elseif (t1 <= 8.5e-113)
		tmp = v * (-t1 / (u * u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(1.0 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -3.1e-145], t$95$1, If[LessEqual[t1, 8.5e-113], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.1e-145 or 8.4999999999999995e-113 < t1

   1. Initial program 63.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-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 99.9

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

   4. Applied rewrites 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. Applied rewrites 77.5%

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

   if -3.1e-145 < t1 < 8.4999999999999995e-113

   1. Initial program 81.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 82.4

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

   4. Applied rewrites 82.4%

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

   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 81.2

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

   7. Applied rewrites 81.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;-t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (* 1.0 (/ v (+ t1 u))))))
   (if (<= t1 -3.1e-145)
     t_1
     (if (<= t1 8.5e-113) (- (* t1 (/ v (* u u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -3.1e-145) {
		tmp = t_1;
	} else if (t1 <= 8.5e-113) {
		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 = -(1.0d0 * (v / (t1 + u)))
    if (t1 <= (-3.1d-145)) then
        tmp = t_1
    else if (t1 <= 8.5d-113) 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 = -(1.0 * (v / (t1 + u)));
	double tmp;
	if (t1 <= -3.1e-145) {
		tmp = t_1;
	} else if (t1 <= 8.5e-113) {
		tmp = -(t1 * (v / (u * u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(1.0 * (v / (t1 + u)))
	tmp = 0
	if t1 <= -3.1e-145:
		tmp = t_1
	elif t1 <= 8.5e-113:
		tmp = -(t1 * (v / (u * u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(1.0 * Float64(v / Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3.1e-145)
		tmp = t_1;
	elseif (t1 <= 8.5e-113)
		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 = -(1.0 * (v / (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3.1e-145)
		tmp = t_1;
	elseif (t1 <= 8.5e-113)
		tmp = -(t1 * (v / (u * u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(1.0 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -3.1e-145], t$95$1, If[LessEqual[t1, 8.5e-113], (-N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.1e-145 or 8.4999999999999995e-113 < t1

   1. Initial program 63.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-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 99.9

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

   4. Applied rewrites 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. Applied rewrites 77.5%

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

   if -3.1e-145 < t1 < 8.4999999999999995e-113

   1. Initial program 81.8%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      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 76.6

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

   5. Applied rewrites 76.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;-t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% 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 69.3%

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

   1. lift-/.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 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 9: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(-\frac{v}{u}\right)\\ \mathbf{if}\;u \leq -2.6 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+179}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* 1.0 (- (/ v u)))))
   (if (<= u -2.6e+87) t_1 (if (<= u 2.4e+179) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 1.0 * -(v / u);
	double tmp;
	if (u <= -2.6e+87) {
		tmp = t_1;
	} else if (u <= 2.4e+179) {
		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 = 1.0d0 * -(v / u)
    if (u <= (-2.6d+87)) then
        tmp = t_1
    else if (u <= 2.4d+179) 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 = 1.0 * -(v / u);
	double tmp;
	if (u <= -2.6e+87) {
		tmp = t_1;
	} else if (u <= 2.4e+179) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 1.0 * -(v / u)
	tmp = 0
	if u <= -2.6e+87:
		tmp = t_1
	elif u <= 2.4e+179:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(1.0 * Float64(-Float64(v / u)))
	tmp = 0.0
	if (u <= -2.6e+87)
		tmp = t_1;
	elseif (u <= 2.4e+179)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 1.0 * -(v / u);
	tmp = 0.0;
	if (u <= -2.6e+87)
		tmp = t_1;
	elseif (u <= 2.4e+179)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(1.0 * (-N[(v / u), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[u, -2.6e+87], t$95$1, If[LessEqual[u, 2.4e+179], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.59999999999999998e87 or 2.40000000000000013e179 < u

   1. Initial program 78.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.9

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in t1 around inf

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

   10. Applied rewrites 35.2%

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

   if -2.59999999999999998e87 < u < 2.40000000000000013e179

   1. Initial program 65.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+87}:\\ \;\;\;\;1 \cdot \left(-\frac{v}{u}\right)\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+179}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-\frac{v}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.3%

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

   1. lift-/.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 98.7

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

4. Applied rewrites 98.7%

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

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. Applied rewrites 60.7%

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

8. Final simplification 60.7%

   \[\leadsto -1 \cdot \frac{v}{t1 + u} \]
9. Add Preprocessing

Alternative 11: 53.2% 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(v / Float64(-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 69.3%

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

3. Taylor expanded in t1 around inf

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

   1. 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 54.0

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

5. Applied rewrites 54.0%

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

6. Final simplification 54.0%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))