Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.6%

Time: 10.5s

Alternatives: 10

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% 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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t1 \cdot v\_m}{\left(-\left(u + t1\right)\right) \cdot \left(u + t1\right)} \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v\_m}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u + t1} \cdot \frac{v\_m}{u + t1}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (<= (/ (* t1 v_m) (* (- (+ u t1)) (+ u t1))) -5e+68)
    (/ (- v_m) (fma (+ 2.0 (/ u t1)) u t1))
    (* (/ (- t1) (+ u t1)) (/ v_m (+ u t1))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (((t1 * v_m) / (-(u + t1) * (u + t1))) <= -5e+68) {
		tmp = -v_m / fma((2.0 + (u / t1)), u, t1);
	} else {
		tmp = (-t1 / (u + t1)) * (v_m / (u + t1));
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if (Float64(Float64(t1 * v_m) / Float64(Float64(-Float64(u + t1)) * Float64(u + t1))) <= -5e+68)
		tmp = Float64(Float64(-v_m) / fma(Float64(2.0 + Float64(u / t1)), u, t1));
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u + t1)) * Float64(v_m / Float64(u + t1)));
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[LessEqual[N[(N[(t1 * v$95$m), $MachinePrecision] / N[((-N[(u + t1), $MachinePrecision]) * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+68], N[((-v$95$m) / N[(N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] * u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(v$95$m / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < -5.0000000000000004e68

   1. Initial program 76.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.6

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -5.0000000000000004e68 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 98.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t1 \cdot v}{\left(-\left(u + t1\right)\right) \cdot \left(u + t1\right)} \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_1 := \frac{-v\_m}{\mathsf{fma}\left(2, u, t1\right)}\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{-t1}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot v\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (let* ((t_1 (/ (- v_m) (fma 2.0 u t1))))
   (*
    v_s
    (if (<= t1 -8e+141)
      t_1
      (if (<= t1 6e+144) (* (/ (- t1) (* (+ u t1) (+ u t1))) v_m) t_1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double t_1 = -v_m / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -8e+141) {
		tmp = t_1;
	} else if (t1 <= 6e+144) {
		tmp = (-t1 / ((u + t1) * (u + t1))) * v_m;
	} else {
		tmp = t_1;
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	t_1 = Float64(Float64(-v_m) / fma(2.0, u, t1))
	tmp = 0.0
	if (t1 <= -8e+141)
		tmp = t_1;
	elseif (t1 <= 6e+144)
		tmp = Float64(Float64(Float64(-t1) / Float64(Float64(u + t1) * Float64(u + t1))) * v_m);
	else
		tmp = t_1;
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := Block[{t$95$1 = N[((-v$95$m) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]}, N[(v$95$s * If[LessEqual[t1, -8e+141], t$95$1, If[LessEqual[t1, 6e+144], N[(N[((-t1) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * v$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.00000000000000014e141 or 5.9999999999999998e144 < t1

   1. Initial program 42.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.4

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

   9. Applied rewrites 92.4%

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

   if -8.00000000000000014e141 < t1 < 5.9999999999999998e144

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 87.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 87.6

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

   4. Applied rewrites 87.6%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))