Rosa's DopplerBench

Percentage Accurate: 73.1% → 97.9%

Time: 7.0s

Alternatives: 9

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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 69.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   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

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

   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 99.4

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

   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 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{-225}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
        (t_2 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -1.9e+48)
     t_2
     (if (<= t1 -5e-138)
       t_1
       (if (<= t1 5.2e-225)
         (* (/ (- t1) u) (/ v u))
         (if (<= t1 4.4e+54) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = -v / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -1.9e+48) {
		tmp = t_2;
	} else if (t1 <= -5e-138) {
		tmp = t_1;
	} else if (t1 <= 5.2e-225) {
		tmp = (-t1 / u) * (v / u);
	} else if (t1 <= 4.4e+54) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	t_2 = Float64(Float64(-v) / fma(2.0, u, t1))
	tmp = 0.0
	if (t1 <= -1.9e+48)
		tmp = t_2;
	elseif (t1 <= -5e-138)
		tmp = t_1;
	elseif (t1 <= 5.2e-225)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	elseif (t1 <= 4.4e+54)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.9e+48], t$95$2, If[LessEqual[t1, -5e-138], t$95$1, If[LessEqual[t1, 5.2e-225], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.4e+54], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.9e48 or 4.3999999999999998e54 < t1

   1. Initial program 55.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      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

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

      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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 95.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 95.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 87.5

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

   9. Applied rewrites 87.5%

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

   if -1.9e48 < t1 < -4.99999999999999989e-138 or 5.20000000000000027e-225 < t1 < 4.3999999999999998e54

   1. Initial program 89.2%

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

   if -4.99999999999999989e-138 < t1 < 5.20000000000000027e-225

   1. Initial program 73.4%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{-225}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.2e-108) (not (<= t1 2.4e-158)))
   (/ (- v) (fma 2.0 u t1))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e-108) || !(t1 <= 2.4e-158)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.2e-108) || !(t1 <= 2.4e-158))
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.2e-108], N[Not[LessEqual[t1, 2.4e-158]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.19999999999999968e-108 or 2.40000000000000007e-158 < t1

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      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

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 94.4

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 94.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 94.4

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.4

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

   9. Applied rewrites 78.4%

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

   if -5.19999999999999968e-108 < t1 < 2.40000000000000007e-158

   1. Initial program 78.5%

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

   3. Taylor expanded in u around 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 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.2e-108) (not (<= t1 2.4e-158)))
   (/ (- v) (fma 2.0 u t1))
   (* v (/ (/ (- t1) u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e-108) || !(t1 <= 2.4e-158)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.2e-108) || !(t1 <= 2.4e-158))
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	else
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.2e-108], N[Not[LessEqual[t1, 2.4e-158]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.19999999999999968e-108 or 2.40000000000000007e-158 < t1

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      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

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 94.4

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 94.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 94.4

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.4

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

   9. Applied rewrites 78.4%

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

   if -5.19999999999999968e-108 < t1 < 2.40000000000000007e-158

   1. Initial program 78.5%

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

   3. Taylor expanded in u around 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 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 82.8%

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

   11. Applied rewrites 89.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 8.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(\frac{u}{t1} + 2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t1}{u} \cdot v\right) \cdot \frac{-1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 8.6e+147)
   (/ (- v) (fma (+ (/ u t1) 2.0) u t1))
   (* (* (/ t1 u) v) (/ -1.0 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8.6e+147) {
		tmp = -v / fma(((u / t1) + 2.0), u, t1);
	} else {
		tmp = ((t1 / u) * v) * (-1.0 / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 8.6e+147)
		tmp = Float64(Float64(-v) / fma(Float64(Float64(u / t1) + 2.0), u, t1));
	else
		tmp = Float64(Float64(Float64(t1 / u) * v) * Float64(-1.0 / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 8.6e+147], N[((-v) / N[(N[(N[(u / t1), $MachinePrecision] + 2.0), $MachinePrecision] * u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 8.5999999999999997e147

   1. Initial program 71.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      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

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

      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 99.4

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 96.6

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

   6. Applied rewrites 96.6%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 96.6

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

   9. Applied rewrites 96.6%

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

   if 8.5999999999999997e147 < u

   1. Initial program 57.4%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 93.6%

      \[\leadsto \left(\frac{t1}{u} \cdot v\right) \cdot \color{blue}{\frac{-1}{u}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.2e-108) (not (<= t1 2.4e-158)))
   (/ (- v) (fma 2.0 u t1))
   (* t1 (/ v (* (- u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e-108) || !(t1 <= 2.4e-158)) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = t1 * (v / (-u * u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.2e-108) || !(t1 <= 2.4e-158))
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	else
		tmp = Float64(t1 * Float64(v / Float64(Float64(-u) * u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.2e-108], N[Not[LessEqual[t1, 2.4e-158]], $MachinePrecision]], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.19999999999999968e-108 or 2.40000000000000007e-158 < t1

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      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

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 94.4

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 94.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 94.4

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.4

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

   9. Applied rewrites 78.4%

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

   if -5.19999999999999968e-108 < t1 < 2.40000000000000007e-158

   1. Initial program 78.5%

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

   3. Taylor expanded in u around 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 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 83.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{\mathsf{fma}\left(2, u, t1\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (- v) (fma 2.0 u t1)))

C

double code(double u, double v, double t1) {
	return -v / fma(2.0, u, t1);
}

Julia

function code(u, v, t1)
	return Float64(Float64(-v) / fma(2.0, u, t1))
end

Wolfram

code[u_, v_, t1_] := N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   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

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

   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 99.4

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

   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 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 94.4

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 94.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 94.4

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

6. Applied rewrites 94.4%

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

7. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 64.4

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

9. Applied rewrites 64.4%

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

Alternative 8: 61.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   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

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

   14. lower-/.f64 98.2

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 63.5

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

7. Applied rewrites 63.5%

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

Alternative 9: 54.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 59.0

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

5. Applied rewrites 59.0%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))