Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.0%

Time: 6.7s

Alternatives: 14

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: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 72.0

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

5. Applied rewrites 72.0%

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

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

Alternative 2: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 7 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right) \cdot \left(-v\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+154)
   (/ (* (/ t1 u) (- v)) (- u t1))
   (if (or (<= u -8e-137) (not (<= u 7e-135)))
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (* (fma (/ u t1) -2.0 1.0) (- v)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+154) {
		tmp = ((t1 / u) * -v) / (u - t1);
	} else if ((u <= -8e-137) || !(u <= 7e-135)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = (fma((u / t1), -2.0, 1.0) * -v) / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+154)
		tmp = Float64(Float64(Float64(t1 / u) * Float64(-v)) / Float64(u - t1));
	elseif ((u <= -8e-137) || !(u <= 7e-135))
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(fma(Float64(u / t1), -2.0, 1.0) * Float64(-v)) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+154], N[(N[(N[(t1 / u), $MachinePrecision] * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -8e-137], N[Not[LessEqual[u, 7e-135]], $MachinePrecision]], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(u / t1), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] * (-v)), $MachinePrecision] / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.15e154

   1. Initial program 61.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 92.4

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

   7. Applied rewrites 92.4%

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

   if -1.15e154 < u < -7.99999999999999982e-137 or 6.9999999999999997e-135 < u

   1. Initial program 85.6%

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

   if -7.99999999999999982e-137 < u < 6.9999999999999997e-135

   1. Initial program 54.3%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 87.5%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right) \cdot \left(-v\right)}{t1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 7 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right) \cdot \left(-v\right)}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 1.16 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+154)
   (/ (* (/ t1 u) (- v)) (- u t1))
   (if (or (<= u -8e-137) (not (<= u 1.16e-133)))
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+154) {
		tmp = ((t1 / u) * -v) / (u - t1);
	} else if ((u <= -8e-137) || !(u <= 1.16e-133)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -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 (u <= (-1.15d+154)) then
        tmp = ((t1 / u) * -v) / (u - t1)
    else if ((u <= (-8d-137)) .or. (.not. (u <= 1.16d-133))) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+154) {
		tmp = ((t1 / u) * -v) / (u - t1);
	} else if ((u <= -8e-137) || !(u <= 1.16e-133)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e+154:
		tmp = ((t1 / u) * -v) / (u - t1)
	elif (u <= -8e-137) or not (u <= 1.16e-133):
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+154)
		tmp = Float64(Float64(Float64(t1 / u) * Float64(-v)) / Float64(u - t1));
	elseif ((u <= -8e-137) || !(u <= 1.16e-133))
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e+154)
		tmp = ((t1 / u) * -v) / (u - t1);
	elseif ((u <= -8e-137) || ~((u <= 1.16e-133)))
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+154], N[(N[(N[(t1 / u), $MachinePrecision] * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -8e-137], N[Not[LessEqual[u, 1.16e-133]], $MachinePrecision]], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.15e154

   1. Initial program 61.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 92.4

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

   7. Applied rewrites 92.4%

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

   if -1.15e154 < u < -7.99999999999999982e-137 or 1.15999999999999997e-133 < u

   1. Initial program 85.6%

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

   if -7.99999999999999982e-137 < u < 1.15999999999999997e-133

   1. Initial program 54.3%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 1.16 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+153}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u - t1}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 1.16 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8e+153)
   (/ (* t1 (/ (- v) u)) (- u t1))
   (if (or (<= u -8e-137) (not (<= u 1.16e-133)))
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8e+153) {
		tmp = (t1 * (-v / u)) / (u - t1);
	} else if ((u <= -8e-137) || !(u <= 1.16e-133)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -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 (u <= (-8d+153)) then
        tmp = (t1 * (-v / u)) / (u - t1)
    else if ((u <= (-8d-137)) .or. (.not. (u <= 1.16d-133))) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8e+153) {
		tmp = (t1 * (-v / u)) / (u - t1);
	} else if ((u <= -8e-137) || !(u <= 1.16e-133)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8e+153:
		tmp = (t1 * (-v / u)) / (u - t1)
	elif (u <= -8e-137) or not (u <= 1.16e-133):
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8e+153)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / Float64(u - t1));
	elseif ((u <= -8e-137) || !(u <= 1.16e-133))
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8e+153)
		tmp = (t1 * (-v / u)) / (u - t1);
	elseif ((u <= -8e-137) || ~((u <= 1.16e-133)))
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8e+153], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -8e-137], N[Not[LessEqual[u, 1.16e-133]], $MachinePrecision]], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8e153

   1. Initial program 61.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if -8e153 < u < -7.99999999999999982e-137 or 1.15999999999999997e-133 < u

   1. Initial program 85.6%

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

   if -7.99999999999999982e-137 < u < 1.15999999999999997e-133

   1. Initial program 54.3%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+153}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u - t1}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 1.16 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 1.16 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.2e+133)
   (* (/ v u) (/ (- t1) u))
   (if (or (<= u -8e-137) (not (<= u 1.16e-133)))
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.2e+133) {
		tmp = (v / u) * (-t1 / u);
	} else if ((u <= -8e-137) || !(u <= 1.16e-133)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -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 (u <= (-7.2d+133)) then
        tmp = (v / u) * (-t1 / u)
    else if ((u <= (-8d-137)) .or. (.not. (u <= 1.16d-133))) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.2e+133) {
		tmp = (v / u) * (-t1 / u);
	} else if ((u <= -8e-137) || !(u <= 1.16e-133)) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.2e+133:
		tmp = (v / u) * (-t1 / u)
	elif (u <= -8e-137) or not (u <= 1.16e-133):
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.2e+133)
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	elseif ((u <= -8e-137) || !(u <= 1.16e-133))
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.2e+133)
		tmp = (v / u) * (-t1 / u);
	elseif ((u <= -8e-137) || ~((u <= 1.16e-133)))
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.2e+133], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -8e-137], N[Not[LessEqual[u, 1.16e-133]], $MachinePrecision]], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.19999999999999956e133

   1. Initial program 63.2%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if -7.19999999999999956e133 < u < -7.99999999999999982e-137 or 1.15999999999999997e-133 < u

   1. Initial program 85.8%

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

   if -7.99999999999999982e-137 < u < 1.15999999999999997e-133

   1. Initial program 54.3%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{-137} \lor \neg \left(u \leq 1.16 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.4e+70)
   (/ (- v) (+ u t1))
   (if (<= t1 3.2e+109)
     (* (/ (- v) (* (+ u t1) (+ u t1))) t1)
     (/ (* -1.0 v) (+ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.4e+70) {
		tmp = -v / (u + t1);
	} else if (t1 <= 3.2e+109) {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	} else {
		tmp = (-1.0 * v) / (-u + 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 <= (-5.4d+70)) then
        tmp = -v / (u + t1)
    else if (t1 <= 3.2d+109) then
        tmp = (-v / ((u + t1) * (u + t1))) * t1
    else
        tmp = ((-1.0d0) * v) / (-u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.4e+70) {
		tmp = -v / (u + t1);
	} else if (t1 <= 3.2e+109) {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.4e+70:
		tmp = -v / (u + t1)
	elif t1 <= 3.2e+109:
		tmp = (-v / ((u + t1) * (u + t1))) * t1
	else:
		tmp = (-1.0 * v) / (-u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.4e+70)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	elseif (t1 <= 3.2e+109)
		tmp = Float64(Float64(Float64(-v) / Float64(Float64(u + t1) * Float64(u + t1))) * t1);
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.4e+70)
		tmp = -v / (u + t1);
	elseif (t1 <= 3.2e+109)
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	else
		tmp = (-1.0 * v) / (-u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.4e+70], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.2e+109], N[(N[((-v) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.3999999999999999e70

   1. Initial program 57.5%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 83.4%

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

   if -5.3999999999999999e70 < t1 < 3.2000000000000001e109

   1. Initial program 82.8%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 82.0%

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

   if 3.2000000000000001e109 < t1

   1. Initial program 55.9%

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 88.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-90} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e-90) (not (<= t1 6.5e-44)))
   (/ (- v) (+ u t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.5d-90)) .or. (.not. (t1 <= 6.5d-44))) then
        tmp = -v / (u + t1)
    else
        tmp = (-t1 * v) / (u * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e-90) or not (t1 <= 6.5e-44):
		tmp = -v / (u + t1)
	else:
		tmp = (-t1 * v) / (u * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.5e-90) || ~((t1 <= 6.5e-44)))
		tmp = -v / (u + t1);
	else
		tmp = (-t1 * v) / (u * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.5e-90], N[Not[LessEqual[t1, 6.5e-44]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.5000000000000001e-90 or 6.5e-44 < t1

   1. Initial program 69.5%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 77.5%

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

   if -1.5000000000000001e-90 < t1 < 6.5e-44

   1. Initial program 79.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-90} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-90} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e-90) (not (<= t1 6.5e-44)))
   (/ (- v) (+ u t1))
   (* v (/ (- t1) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44)) {
		tmp = -v / (u + t1);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.5d-90)) .or. (.not. (t1 <= 6.5d-44))) then
        tmp = -v / (u + t1)
    else
        tmp = v * (-t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44)) {
		tmp = -v / (u + t1);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e-90) or not (t1 <= 6.5e-44):
		tmp = -v / (u + t1)
	else:
		tmp = v * (-t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.5e-90) || ~((t1 <= 6.5e-44)))
		tmp = -v / (u + t1);
	else
		tmp = v * (-t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.5e-90], N[Not[LessEqual[t1, 6.5e-44]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.5000000000000001e-90 or 6.5e-44 < t1

   1. Initial program 69.5%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 77.5%

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

   if -1.5000000000000001e-90 < t1 < 6.5e-44

   1. Initial program 79.4%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.3

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 73.0

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

   6. Applied rewrites 73.0%

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

   7. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.6

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

   9. Applied rewrites 74.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-90} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-90} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e-90) (not (<= t1 6.5e-44)))
   (/ (- v) (+ u t1))
   (* (- t1) (/ v (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44)) {
		tmp = -v / (u + t1);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.5d-90)) .or. (.not. (t1 <= 6.5d-44))) then
        tmp = -v / (u + t1)
    else
        tmp = -t1 * (v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44)) {
		tmp = -v / (u + t1);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e-90) or not (t1 <= 6.5e-44):
		tmp = -v / (u + t1)
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.5e-90) || !(t1 <= 6.5e-44))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.5e-90) || ~((t1 <= 6.5e-44)))
		tmp = -v / (u + t1);
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.5e-90], N[Not[LessEqual[t1, 6.5e-44]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.5000000000000001e-90 or 6.5e-44 < t1

   1. Initial program 69.5%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 77.5%

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

   if -1.5000000000000001e-90 < t1 < 6.5e-44

   1. Initial program 79.4%

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   6. 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-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.1

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

   7. Applied rewrites 72.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-90} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 72.0

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

5. Applied rewrites 72.0%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 11: 60.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (-1.0 * v) / (-u + 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 = ((-1.0d0) * v) / (-u + t1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 58.4%

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

8. Final simplification 58.4%

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

Alternative 12: 60.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 72.0

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

5. Applied rewrites 72.0%

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 58.4%

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

Alternative 13: 53.1% 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 73.3%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
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 52.2

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

5. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\frac{-v}{t1}} \]
6. Add Preprocessing

Alternative 14: 14.2% accurate, 2.5× 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 / 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 73.3%

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

4. Applied rewrites 55.0%

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

5. Taylor expanded in u around 0

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

   1. lower-/.f64 15.9

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

7. Applied rewrites 15.9%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))