Rosa's DopplerBench

Percentage Accurate: 72.1% → 97.9%

Time: 6.6s

Alternatives: 8

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: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-2neg N/A

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

   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

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 98.6

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{+134}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{u}{t1} - 1\right) \cdot v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.5e+70)
   (/ (- v) (+ t1 u))
   (if (<= t1 1.35e+134)
     (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))
     (/ (* (- (/ u t1) 1.0) v) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.5e+70) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 1.35e+134) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (((u / t1) - 1.0) * v) / (t1 + 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 <= (-4.5d+70)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 1.35d+134) then
        tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
    else
        tmp = (((u / t1) - 1.0d0) * v) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.5e+70) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 1.35e+134) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (((u / t1) - 1.0) * v) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.5e+70:
		tmp = -v / (t1 + u)
	elif t1 <= 1.35e+134:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = (((u / t1) - 1.0) * v) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.5e+70)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 1.35e+134)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(Float64(Float64(Float64(u / t1) - 1.0) * v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.5e+70)
		tmp = -v / (t1 + u);
	elseif (t1 <= 1.35e+134)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = (((u / t1) - 1.0) * v) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.5e+70], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.35e+134], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.4999999999999999e70

   1. Initial program 47.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   9. Applied rewrites 98.0%

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

   if -4.4999999999999999e70 < t1 < 1.35e134

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 97.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t1 around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 97.9

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

   7. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if 1.35e134 < t1

   1. Initial program 30.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 94.3

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

   9. Applied rewrites 94.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{+134}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{u}{t1} - 1\right) \cdot v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+70} \lor \neg \left(t1 \leq 1.55 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e+70) (not (<= t1 1.55e+120)))
   (/ (- v) (+ t1 u))
   (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+70) || !(t1 <= 1.55e+120)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * (-t1 / ((t1 + u) * (t1 + 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 <= (-4.5d+70)) .or. (.not. (t1 <= 1.55d+120))) then
        tmp = -v / (t1 + u)
    else
        tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+70) || !(t1 <= 1.55e+120)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.5e+70) or not (t1 <= 1.55e+120):
		tmp = -v / (t1 + u)
	else:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e+70) || !(t1 <= 1.55e+120))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.5e+70) || ~((t1 <= 1.55e+120)))
		tmp = -v / (t1 + u);
	else
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e+70], N[Not[LessEqual[t1, 1.55e+120]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.4999999999999999e70 or 1.54999999999999987e120 < t1

   1. Initial program 43.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   9. Applied rewrites 95.6%

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

   if -4.4999999999999999e70 < t1 < 1.54999999999999987e120

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 97.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in t1 around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 88.7

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

   9. Applied rewrites 88.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+70} \lor \neg \left(t1 \leq 1.55 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-62} \lor \neg \left(t1 \leq 8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.9e-62) (not (<= t1 8e+14)))
   (/ (- v) (+ t1 u))
   (/ (* (/ (- v) u) t1) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.9e-62) || !(t1 <= 8e+14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = ((-v / u) * t1) / 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 <= (-1.9d-62)) .or. (.not. (t1 <= 8d+14))) then
        tmp = -v / (t1 + u)
    else
        tmp = ((-v / u) * t1) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.9e-62) || !(t1 <= 8e+14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = ((-v / u) * t1) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.9e-62) or not (t1 <= 8e+14):
		tmp = -v / (t1 + u)
	else:
		tmp = ((-v / u) * t1) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.9e-62) || !(t1 <= 8e+14))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.9e-62) || ~((t1 <= 8e+14)))
		tmp = -v / (t1 + u);
	else
		tmp = ((-v / u) * t1) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.9e-62], N[Not[LessEqual[t1, 8e+14]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.90000000000000003e-62 or 8e14 < t1

   1. Initial program 60.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if -1.90000000000000003e-62 < t1 < 8e14

   1. Initial program 79.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 82.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-62} \lor \neg \left(t1 \leq 8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-62} \lor \neg \left(t1 \leq 8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.9e-62) (not (<= t1 8e+14)))
   (/ (- v) (+ t1 u))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.9e-62) || !(t1 <= 8e+14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.9d-62)) .or. (.not. (t1 <= 8d+14))) then
        tmp = -v / (t1 + u)
    else
        tmp = (t1 / u) * (-v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.9e-62) || !(t1 <= 8e+14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.9e-62) or not (t1 <= 8e+14):
		tmp = -v / (t1 + u)
	else:
		tmp = (t1 / u) * (-v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.9e-62) || !(t1 <= 8e+14))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.9e-62) || ~((t1 <= 8e+14)))
		tmp = -v / (t1 + u);
	else
		tmp = (t1 / u) * (-v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.9e-62], N[Not[LessEqual[t1, 8e+14]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.90000000000000003e-62 or 8e14 < t1

   1. Initial program 60.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if -1.90000000000000003e-62 < t1 < 8e14

   1. Initial program 79.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-62} \lor \neg \left(t1 \leq 8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-63} \lor \neg \left(t1 \leq 5.5 \cdot 10^{-136}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e-63) (not (<= t1 5.5e-136)))
   (/ (- v) (+ t1 u))
   (* t1 (/ v (* (- u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-63) || !(t1 <= 5.5e-136)) {
		tmp = -v / (t1 + u);
	} 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 <= (-1d-63)) .or. (.not. (t1 <= 5.5d-136))) then
        tmp = -v / (t1 + u)
    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 <= -1e-63) || !(t1 <= 5.5e-136)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * (v / (-u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e-63) or not (t1 <= 5.5e-136):
		tmp = -v / (t1 + u)
	else:
		tmp = t1 * (v / (-u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e-63) || !(t1 <= 5.5e-136))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 * Float64(v / Float64(Float64(-u) * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e-63) || ~((t1 <= 5.5e-136)))
		tmp = -v / (t1 + u);
	else
		tmp = t1 * (v / (-u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e-63], N[Not[LessEqual[t1, 5.5e-136]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000007e-63 or 5.4999999999999999e-136 < t1

   1. Initial program 63.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. times-frac N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      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

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.1

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

   9. Applied rewrites 86.1%

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

   if -1.00000000000000007e-63 < t1 < 5.4999999999999999e-136

   1. Initial program 79.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 80.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-63} \lor \neg \left(t1 \leq 5.5 \cdot 10^{-136}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-2neg N/A

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

   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

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 98.6

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 66.2

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

9. Applied rewrites 66.2%

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

Alternative 8: 54.2% 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(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 69.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 56.8

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

5. Applied rewrites 56.8%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))