Rosa's DopplerBench

Percentage Accurate: 72.6% → 97.8%

Time: 10.2s

Alternatives: 12

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: 72.6% 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: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. lift-neg.f64 N/A

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

   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. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

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

5. Final simplification 97.5%

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

Alternative 2: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u \cdot -2, v\right)}{-t1}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (* v (/ t1 (* (+ t1 u) t_1)))))
   (if (<= u -1.75e+130)
     (/ (* t1 (/ (- v) u)) u)
     (if (<= u -5.6e-156)
       t_2
       (if (<= u 7.6e-156)
         (/ (fma (/ v t1) (* u -2.0) v) (- t1))
         (if (<= u 1.6e+80) t_2 (* (/ v t_1) (/ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v * (t1 / ((t1 + u) * t_1));
	double tmp;
	if (u <= -1.75e+130) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -5.6e-156) {
		tmp = t_2;
	} else if (u <= 7.6e-156) {
		tmp = fma((v / t1), (u * -2.0), v) / -t1;
	} else if (u <= 1.6e+80) {
		tmp = t_2;
	} else {
		tmp = (v / t_1) * (t1 / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)))
	tmp = 0.0
	if (u <= -1.75e+130)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= -5.6e-156)
		tmp = t_2;
	elseif (u <= 7.6e-156)
		tmp = Float64(fma(Float64(v / t1), Float64(u * -2.0), v) / Float64(-t1));
	elseif (u <= 1.6e+80)
		tmp = t_2;
	else
		tmp = Float64(Float64(v / t_1) * Float64(t1 / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.75e+130], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, -5.6e-156], t$95$2, If[LessEqual[u, 7.6e-156], N[(N[(N[(v / t1), $MachinePrecision] * N[(u * -2.0), $MachinePrecision] + v), $MachinePrecision] / (-t1)), $MachinePrecision], If[LessEqual[u, 1.6e+80], t$95$2, N[(N[(v / t$95$1), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.75e130

   1. Initial program 70.7%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if -1.75e130 < u < -5.6000000000000003e-156 or 7.60000000000000015e-156 < u < 1.59999999999999995e80

   1. Initial program 78.7%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -5.6000000000000003e-156 < u < 7.60000000000000015e-156

   1. Initial program 61.1%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. distribute-neg-in N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if 1.59999999999999995e80 < u

   1. Initial program 79.0%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t1 around 0

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

      1. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u \cdot -2, v\right)}{-t1}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{\mathsf{fma}\left(u, \frac{2}{t1}, -1\right)}{t1}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (* v (/ t1 (* (+ t1 u) t_1)))))
   (if (<= u -1.75e+130)
     (/ (* t1 (/ (- v) u)) u)
     (if (<= u -5.6e-156)
       t_2
       (if (<= u 7.6e-156)
         (* v (/ (fma u (/ 2.0 t1) -1.0) t1))
         (if (<= u 1.6e+80) t_2 (* (/ v t_1) (/ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v * (t1 / ((t1 + u) * t_1));
	double tmp;
	if (u <= -1.75e+130) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -5.6e-156) {
		tmp = t_2;
	} else if (u <= 7.6e-156) {
		tmp = v * (fma(u, (2.0 / t1), -1.0) / t1);
	} else if (u <= 1.6e+80) {
		tmp = t_2;
	} else {
		tmp = (v / t_1) * (t1 / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)))
	tmp = 0.0
	if (u <= -1.75e+130)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= -5.6e-156)
		tmp = t_2;
	elseif (u <= 7.6e-156)
		tmp = Float64(v * Float64(fma(u, Float64(2.0 / t1), -1.0) / t1));
	elseif (u <= 1.6e+80)
		tmp = t_2;
	else
		tmp = Float64(Float64(v / t_1) * Float64(t1 / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.75e+130], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, -5.6e-156], t$95$2, If[LessEqual[u, 7.6e-156], N[(v * N[(N[(u * N[(2.0 / t1), $MachinePrecision] + -1.0), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.6e+80], t$95$2, N[(N[(v / t$95$1), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.75e130

   1. Initial program 70.7%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if -1.75e130 < u < -5.6000000000000003e-156 or 7.60000000000000015e-156 < u < 1.59999999999999995e80

   1. Initial program 78.7%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -5.6000000000000003e-156 < u < 7.60000000000000015e-156

   1. Initial program 61.1%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 65.3

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

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

   if 1.59999999999999995e80 < u

   1. Initial program 79.0%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t1 around 0

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

      1. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{\mathsf{fma}\left(u, \frac{2}{t1}, -1\right)}{t1}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (* v (/ t1 (* (+ t1 u) t_1)))))
   (if (<= u -1.75e+130)
     (/ (* t1 (/ (- v) u)) u)
     (if (<= u -5.6e-156)
       t_2
       (if (<= u 7.6e-156)
         (/ (- v) t1)
         (if (<= u 1.6e+80) t_2 (* (/ v t_1) (/ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v * (t1 / ((t1 + u) * t_1));
	double tmp;
	if (u <= -1.75e+130) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -5.6e-156) {
		tmp = t_2;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 1.6e+80) {
		tmp = t_2;
	} else {
		tmp = (v / t_1) * (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = v * (t1 / ((t1 + u) * t_1))
    if (u <= (-1.75d+130)) then
        tmp = (t1 * (-v / u)) / u
    else if (u <= (-5.6d-156)) then
        tmp = t_2
    else if (u <= 7.6d-156) then
        tmp = -v / t1
    else if (u <= 1.6d+80) then
        tmp = t_2
    else
        tmp = (v / t_1) * (t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v * (t1 / ((t1 + u) * t_1));
	double tmp;
	if (u <= -1.75e+130) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -5.6e-156) {
		tmp = t_2;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 1.6e+80) {
		tmp = t_2;
	} else {
		tmp = (v / t_1) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = v * (t1 / ((t1 + u) * t_1))
	tmp = 0
	if u <= -1.75e+130:
		tmp = (t1 * (-v / u)) / u
	elif u <= -5.6e-156:
		tmp = t_2
	elif u <= 7.6e-156:
		tmp = -v / t1
	elif u <= 1.6e+80:
		tmp = t_2
	else:
		tmp = (v / t_1) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)))
	tmp = 0.0
	if (u <= -1.75e+130)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= -5.6e-156)
		tmp = t_2;
	elseif (u <= 7.6e-156)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 1.6e+80)
		tmp = t_2;
	else
		tmp = Float64(Float64(v / t_1) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = v * (t1 / ((t1 + u) * t_1));
	tmp = 0.0;
	if (u <= -1.75e+130)
		tmp = (t1 * (-v / u)) / u;
	elseif (u <= -5.6e-156)
		tmp = t_2;
	elseif (u <= 7.6e-156)
		tmp = -v / t1;
	elseif (u <= 1.6e+80)
		tmp = t_2;
	else
		tmp = (v / t_1) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.75e+130], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, -5.6e-156], t$95$2, If[LessEqual[u, 7.6e-156], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 1.6e+80], t$95$2, N[(N[(v / t$95$1), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.75e130

   1. Initial program 70.7%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if -1.75e130 < u < -5.6000000000000003e-156 or 7.60000000000000015e-156 < u < 1.59999999999999995e80

   1. Initial program 78.7%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -5.6000000000000003e-156 < u < 7.60000000000000015e-156

   1. Initial program 61.1%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 1.59999999999999995e80 < u

   1. Initial program 79.0%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t1 around 0

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

      1. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ t_2 := \frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ t1 (* (+ t1 u) (- (- u) t1)))))
        (t_2 (/ (* t1 (/ (- v) u)) u)))
   (if (<= u -1.75e+130)
     t_2
     (if (<= u -5.6e-156)
       t_1
       (if (<= u 7.6e-156) (/ (- v) t1) (if (<= u 6e+72) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double t_2 = (t1 * (-v / u)) / u;
	double tmp;
	if (u <= -1.75e+130) {
		tmp = t_2;
	} else if (u <= -5.6e-156) {
		tmp = t_1;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 6e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
    t_2 = (t1 * (-v / u)) / u
    if (u <= (-1.75d+130)) then
        tmp = t_2
    else if (u <= (-5.6d-156)) then
        tmp = t_1
    else if (u <= 7.6d-156) then
        tmp = -v / t1
    else if (u <= 6d+72) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double t_2 = (t1 * (-v / u)) / u;
	double tmp;
	if (u <= -1.75e+130) {
		tmp = t_2;
	} else if (u <= -5.6e-156) {
		tmp = t_1;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 6e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
	t_2 = (t1 * (-v / u)) / u
	tmp = 0
	if u <= -1.75e+130:
		tmp = t_2
	elif u <= -5.6e-156:
		tmp = t_1
	elif u <= 7.6e-156:
		tmp = -v / t1
	elif u <= 6e+72:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))))
	t_2 = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u)
	tmp = 0.0
	if (u <= -1.75e+130)
		tmp = t_2;
	elseif (u <= -5.6e-156)
		tmp = t_1;
	elseif (u <= 7.6e-156)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 6e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	t_2 = (t1 * (-v / u)) / u;
	tmp = 0.0;
	if (u <= -1.75e+130)
		tmp = t_2;
	elseif (u <= -5.6e-156)
		tmp = t_1;
	elseif (u <= 7.6e-156)
		tmp = -v / t1;
	elseif (u <= 6e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]}, If[LessEqual[u, -1.75e+130], t$95$2, If[LessEqual[u, -5.6e-156], t$95$1, If[LessEqual[u, 7.6e-156], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 6e+72], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.75e130 or 6.00000000000000006e72 < u

   1. Initial program 74.7%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 92.0

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

   7. Applied rewrites 92.0%

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

   if -1.75e130 < u < -5.6000000000000003e-156 or 7.60000000000000015e-156 < u < 6.00000000000000006e72

   1. Initial program 78.8%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -5.6000000000000003e-156 < u < 7.60000000000000015e-156

   1. Initial program 61.1%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+72}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ t_2 := \frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{if}\;u \leq -9.5 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ t1 (* (+ t1 u) (- (- u) t1)))))
        (t_2 (/ (* v (/ t1 u)) (- u))))
   (if (<= u -9.5e+130)
     t_2
     (if (<= u -5.6e-156)
       t_1
       (if (<= u 7.6e-156) (/ (- v) t1) (if (<= u 5.6e+74) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double t_2 = (v * (t1 / u)) / -u;
	double tmp;
	if (u <= -9.5e+130) {
		tmp = t_2;
	} else if (u <= -5.6e-156) {
		tmp = t_1;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 5.6e+74) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
    t_2 = (v * (t1 / u)) / -u
    if (u <= (-9.5d+130)) then
        tmp = t_2
    else if (u <= (-5.6d-156)) then
        tmp = t_1
    else if (u <= 7.6d-156) then
        tmp = -v / t1
    else if (u <= 5.6d+74) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double t_2 = (v * (t1 / u)) / -u;
	double tmp;
	if (u <= -9.5e+130) {
		tmp = t_2;
	} else if (u <= -5.6e-156) {
		tmp = t_1;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 5.6e+74) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
	t_2 = (v * (t1 / u)) / -u
	tmp = 0
	if u <= -9.5e+130:
		tmp = t_2
	elif u <= -5.6e-156:
		tmp = t_1
	elif u <= 7.6e-156:
		tmp = -v / t1
	elif u <= 5.6e+74:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))))
	t_2 = Float64(Float64(v * Float64(t1 / u)) / Float64(-u))
	tmp = 0.0
	if (u <= -9.5e+130)
		tmp = t_2;
	elseif (u <= -5.6e-156)
		tmp = t_1;
	elseif (u <= 7.6e-156)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 5.6e+74)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	t_2 = (v * (t1 / u)) / -u;
	tmp = 0.0;
	if (u <= -9.5e+130)
		tmp = t_2;
	elseif (u <= -5.6e-156)
		tmp = t_1;
	elseif (u <= 7.6e-156)
		tmp = -v / t1;
	elseif (u <= 5.6e+74)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -9.5e+130], t$95$2, If[LessEqual[u, -5.6e-156], t$95$1, If[LessEqual[u, 7.6e-156], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 5.6e+74], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -9.5000000000000009e130 or 5.60000000000000003e74 < u

   1. Initial program 74.2%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 91.8

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

   7. Applied rewrites 91.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 91.3

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

   9. Applied rewrites 91.3%

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

   if -9.5000000000000009e130 < u < -5.6000000000000003e-156 or 7.60000000000000015e-156 < u < 5.60000000000000003e74

   1. Initial program 79.2%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if -5.6000000000000003e-156 < u < 7.60000000000000015e-156

   1. Initial program 61.1%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+74}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ t_2 := \frac{v}{u} \cdot \left(-\frac{t1}{u}\right)\\ \mathbf{if}\;u \leq -2.45 \cdot 10^{+147}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ t1 (* (+ t1 u) (- (- u) t1)))))
        (t_2 (* (/ v u) (- (/ t1 u)))))
   (if (<= u -2.45e+147)
     t_2
     (if (<= u -5.6e-156)
       t_1
       (if (<= u 7.6e-156) (/ (- v) t1) (if (<= u 6e+72) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double t_2 = (v / u) * -(t1 / u);
	double tmp;
	if (u <= -2.45e+147) {
		tmp = t_2;
	} else if (u <= -5.6e-156) {
		tmp = t_1;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 6e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
    t_2 = (v / u) * -(t1 / u)
    if (u <= (-2.45d+147)) then
        tmp = t_2
    else if (u <= (-5.6d-156)) then
        tmp = t_1
    else if (u <= 7.6d-156) then
        tmp = -v / t1
    else if (u <= 6d+72) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double t_2 = (v / u) * -(t1 / u);
	double tmp;
	if (u <= -2.45e+147) {
		tmp = t_2;
	} else if (u <= -5.6e-156) {
		tmp = t_1;
	} else if (u <= 7.6e-156) {
		tmp = -v / t1;
	} else if (u <= 6e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
	t_2 = (v / u) * -(t1 / u)
	tmp = 0
	if u <= -2.45e+147:
		tmp = t_2
	elif u <= -5.6e-156:
		tmp = t_1
	elif u <= 7.6e-156:
		tmp = -v / t1
	elif u <= 6e+72:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))))
	t_2 = Float64(Float64(v / u) * Float64(-Float64(t1 / u)))
	tmp = 0.0
	if (u <= -2.45e+147)
		tmp = t_2;
	elseif (u <= -5.6e-156)
		tmp = t_1;
	elseif (u <= 7.6e-156)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 6e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	t_2 = (v / u) * -(t1 / u);
	tmp = 0.0;
	if (u <= -2.45e+147)
		tmp = t_2;
	elseif (u <= -5.6e-156)
		tmp = t_1;
	elseif (u <= 7.6e-156)
		tmp = -v / t1;
	elseif (u <= 6e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(v / u), $MachinePrecision] * (-N[(t1 / u), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[u, -2.45e+147], t$95$2, If[LessEqual[u, -5.6e-156], t$95$1, If[LessEqual[u, 7.6e-156], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 6e+72], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.4499999999999999e147 or 6.00000000000000006e72 < u

   1. Initial program 75.3%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. distribute-lft-neg-out N/A

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

      2. *-commutative N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if -2.4499999999999999e147 < u < -5.6000000000000003e-156 or 7.60000000000000015e-156 < u < 6.00000000000000006e72

   1. Initial program 78.2%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.6

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

   4. Applied rewrites 85.6%

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

   if -5.6000000000000003e-156 < u < 7.60000000000000015e-156

   1. Initial program 61.1%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.45 \cdot 10^{+147}:\\ \;\;\;\;\frac{v}{u} \cdot \left(-\frac{t1}{u}\right)\\ \mathbf{elif}\;u \leq -5.6 \cdot 10^{-156}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+72}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \left(-\frac{t1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\frac{v}{t\_1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-1}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -1e+155)
     (/ v t_1)
     (if (<= t1 1.1e+128)
       (* v (/ t1 (* (+ t1 u) t_1)))
       (* v (/ -1.0 (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -1e+155) {
		tmp = v / t_1;
	} else if (t1 <= 1.1e+128) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v * (-1.0 / (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) :: t_1
    real(8) :: tmp
    t_1 = -u - t1
    if (t1 <= (-1d+155)) then
        tmp = v / t_1
    else if (t1 <= 1.1d+128) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = v * ((-1.0d0) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -1e+155) {
		tmp = v / t_1;
	} else if (t1 <= 1.1e+128) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v * (-1.0 / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -1e+155:
		tmp = v / t_1
	elif t1 <= 1.1e+128:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = v * (-1.0 / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -1e+155)
		tmp = Float64(v / t_1);
	elseif (t1 <= 1.1e+128)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = Float64(v * Float64(-1.0 / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -1e+155)
		tmp = v / t_1;
	elseif (t1 <= 1.1e+128)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = v * (-1.0 / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -1e+155], N[(v / t$95$1), $MachinePrecision], If[LessEqual[t1, 1.1e+128], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(-1.0 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.00000000000000001e155

   1. Initial program 53.7%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 88.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity 88.5

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

   9. Applied rewrites 88.5%

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

   if -1.00000000000000001e155 < t1 < 1.10000000000000008e128

   1. Initial program 80.9%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.1

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

   4. Applied rewrites 85.1%

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

   if 1.10000000000000008e128 < t1

   1. Initial program 50.1%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 84.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 84.7

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

   9. Applied rewrites 84.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-1}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -6 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.65 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ v (* u u)))))
   (if (<= u -6e+36) t_1 (if (<= u 1.65e+18) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / (u * u));
	double tmp;
	if (u <= -6e+36) {
		tmp = t_1;
	} else if (u <= 1.65e+18) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 * (v / (u * u))
    if (u <= (-6d+36)) then
        tmp = t_1
    else if (u <= 1.65d+18) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / (u * u));
	double tmp;
	if (u <= -6e+36) {
		tmp = t_1;
	} else if (u <= 1.65e+18) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 * (v / (u * u))
	tmp = 0
	if u <= -6e+36:
		tmp = t_1
	elif u <= 1.65e+18:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -6e+36)
		tmp = t_1;
	elseif (u <= 1.65e+18)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 * (v / (u * u));
	tmp = 0.0;
	if (u <= -6e+36)
		tmp = t_1;
	elseif (u <= 1.65e+18)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -6e+36], t$95$1, If[LessEqual[u, 1.65e+18], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -6e36 or 1.65e18 < u

   1. Initial program 79.5%

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

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -6e36 < u < 1.65e18

   1. Initial program 66.3%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 77.9

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

   5. Applied rewrites 77.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.65 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u}\\ \mathbf{if}\;u \leq -4.2 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) u)))
   (if (<= u -4.2e+152) t_1 (if (<= u 3.8e+94) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / u;
	double tmp;
	if (u <= -4.2e+152) {
		tmp = t_1;
	} else if (u <= 3.8e+94) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / u
    if (u <= (-4.2d+152)) then
        tmp = t_1
    else if (u <= 3.8d+94) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / u;
	double tmp;
	if (u <= -4.2e+152) {
		tmp = t_1;
	} else if (u <= 3.8e+94) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / u
	tmp = 0
	if u <= -4.2e+152:
		tmp = t_1
	elif u <= 3.8e+94:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / u)
	tmp = 0.0
	if (u <= -4.2e+152)
		tmp = t_1;
	elseif (u <= 3.8e+94)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / u;
	tmp = 0.0;
	if (u <= -4.2e+152)
		tmp = t_1;
	elseif (u <= 3.8e+94)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / u), $MachinePrecision]}, If[LessEqual[u, -4.2e+152], t$95$1, If[LessEqual[u, 3.8e+94], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -4.2000000000000003e152 or 3.7999999999999996e94 < u

   1. Initial program 75.3%

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

      1. lift-neg.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 33.1

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

   10. Applied rewrites 33.1%

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

   if -4.2000000000000003e152 < u < 3.7999999999999996e94

   1. Initial program 72.0%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.5

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

   5. Applied rewrites 64.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 1.8× speedup

Math

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

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(v / Float64(Float64(-u) - t1))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. lift-neg.f64 N/A

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

   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. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in t1 around inf

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

7. Applied rewrites 54.4%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-rgt-identity 54.4

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

9. Applied rewrites 54.4%

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

10. Final simplification 54.4%

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

Alternative 12: 17.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. lift-neg.f64 N/A

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

   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. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in t1 around inf

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

7. Applied rewrites 54.4%

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

8. Taylor expanded in t1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 14.9

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

10. Applied rewrites 14.9%

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

11. Final simplification 14.9%

    \[\leadsto \frac{-v}{u} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))