Rosa's DopplerBench

Percentage Accurate: 73.2% → 98.2%

Time: 12.3s

Alternatives: 11

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(0 - t1\right) - u} \cdot \frac{t1}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (* (/ v (- (- 0.0 t1) u)) (/ t1 (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (v / ((0.0 - t1) - u)) * (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 / ((0.0d0 - t1) - u)) * (t1 / (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (v / ((0.0 - t1) - u)) * (t1 / (t1 + u));
}

Python

def code(u, v, t1):
	return (v / ((0.0 - t1) - u)) * (t1 / (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(Float64(0.0 - t1) - u)) * Float64(t1 / Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / ((0.0 - t1) - u)) * (t1 / (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

   1. *-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

   10. --lowering--.f64 N/A

       \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{0 - v}{\color{blue}{t1 + u}} \cdot \frac{t1}{t1 + u} \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \frac{0 - v}{t1 + u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

   13. +-lowering-+.f64 98.0

       \[\leadsto \frac{0 - v}{t1 + u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

4. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{0 - v}{t1 + u} \cdot \frac{t1}{t1 + u}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 98.0

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

   \[\leadsto \frac{v}{\left(0 - t1\right) - u} \cdot \frac{t1}{t1 + u} \]
8. Add Preprocessing

Alternative 2: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+114}:\\ \;\;\;\;t1 \cdot \frac{0 - v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (/ v (fma u 2.0 t1)))))
   (if (<= t1 -3.1e+51)
     t_1
     (if (<= t1 9e+114) (* t1 (/ (- 0.0 v) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (v / fma(u, 2.0, t1));
	double tmp;
	if (t1 <= -3.1e+51) {
		tmp = t_1;
	} else if (t1 <= 9e+114) {
		tmp = t1 * ((0.0 - v) / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(v / fma(u, 2.0, t1)))
	tmp = 0.0
	if (t1 <= -3.1e+51)
		tmp = t_1;
	elseif (t1 <= 9e+114)
		tmp = Float64(t1 * Float64(Float64(0.0 - v) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.1e+51], t$95$1, If[LessEqual[t1, 9e+114], N[(t1 * N[(N[(0.0 - v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.10000000000000011e51 or 9.0000000000000001e114 < t1

   1. Initial program 44.7%

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

   3. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 43.3

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

   5. Simplified 43.3%

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

      1. times-frac N/A

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

      2. distribute-frac-neg N/A

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

      3. *-inverses N/A

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

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

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. accelerator-lowering-fma.f64 90.5

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

   7. Applied egg-rr 90.5%

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

   if -3.10000000000000011e51 < t1 < 9.0000000000000001e114

   1. Initial program 81.6%

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 85.6

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

   4. Applied egg-rr 85.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+114}:\\ \;\;\;\;t1 \cdot \frac{0 - v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4 \cdot 10^{+179}:\\ \;\;\;\;\frac{0 - v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(0 - t1\right) - u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 4e+179)
   (/ (- 0.0 v) (fma u (+ 2.0 (/ u t1)) t1))
   (* t1 (/ (/ v (- (- 0.0 t1) u)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 4e+179) {
		tmp = (0.0 - v) / fma(u, (2.0 + (u / t1)), t1);
	} else {
		tmp = t1 * ((v / ((0.0 - t1) - u)) / (t1 + u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 4e+179)
		tmp = Float64(Float64(0.0 - v) / fma(u, Float64(2.0 + Float64(u / t1)), t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(Float64(0.0 - t1) - u)) / Float64(t1 + u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 4e+179], N[(N[(0.0 - v), $MachinePrecision] / N[(u * N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] + t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.99999999999999992e179

   1. Initial program 73.7%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{0 - v}{\color{blue}{t1 + u}} \cdot \frac{t1}{t1 + u} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      13. +-lowering-+.f64 98.2

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{0 - v}{t1 + u} \cdot \frac{t1}{t1 + u}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{0 - v}{t1 + u}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{t1}}} \cdot \frac{0 - v}{t1 + u} \]

      3. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(0 - v\right)}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1 \cdot \left(0 - v\right)\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 - v\right)}\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)} \]

      6. flip3-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)} \]

      7. flip3-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 - v\right)}\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)} \]

      8. sub0-neg N/A

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

      9. remove-double-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.f64 96.1

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in t1 around inf

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

   9. Simplified 96.1%

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

   if 3.99999999999999992e179 < v

   1. Initial program 51.7%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{0 - v}{\color{blue}{t1 + u}} \cdot \frac{t1}{t1 + u} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      13. +-lowering-+.f64 96.3

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

   4. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{\frac{0 - v}{t1 + u} \cdot \frac{t1}{t1 + u}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-frac-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-lowering-*.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 92.5

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

   6. Applied egg-rr 92.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 4 \cdot 10^{+179}:\\ \;\;\;\;\frac{0 - v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(0 - t1\right) - u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118}:\\ \;\;\;\;0 - \frac{t1 \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+118)
   (- 0.0 (/ (* t1 (/ v u)) u))
   (/ (- 0.0 v) (fma u (+ 2.0 (/ u t1)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+118) {
		tmp = 0.0 - ((t1 * (v / u)) / u);
	} else {
		tmp = (0.0 - v) / fma(u, (2.0 + (u / t1)), t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+118)
		tmp = Float64(0.0 - Float64(Float64(t1 * Float64(v / u)) / u));
	else
		tmp = Float64(Float64(0.0 - v) / fma(u, Float64(2.0 + Float64(u / t1)), t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+118], N[(0.0 - N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - v), $MachinePrecision] / N[(u * N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] + t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.80000000000000016e118

   1. Initial program 80.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - t1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{0 - t1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{0 - t1}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \cdot v \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{0 - t1}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \cdot v \]

      10. +-lowering-+.f64 76.3

          \[\leadsto \frac{0 - t1}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \cdot v \]

   4. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{0 - t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

   5. Taylor expanded in t1 around 0

      \[\leadsto \frac{0 - t1}{\color{blue}{{u}^{2}}} \cdot v \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{0 - t1}{\color{blue}{u \cdot u}} \cdot v \]

      2. *-lowering-*.f64 76.3

         \[\leadsto \frac{0 - t1}{\color{blue}{u \cdot u}} \cdot v \]

   7. Simplified 76.3%

      \[\leadsto \frac{0 - t1}{\color{blue}{u \cdot u}} \cdot v \]
   8. Step-by-step derivation

      1. neg-sub0 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. +-lft-identity N/A

         \[\leadsto \frac{\frac{v \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(t1\right)\right)\right)}}{u}}{u} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{v \cdot \color{blue}{\left(\left(\mathsf{neg}\left(t1\right)\right) + 0\right)}}{u}}{u} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{\frac{\color{blue}{v \cdot \left(\mathsf{neg}\left(t1\right)\right) + v \cdot 0}}{u}}{u} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{\frac{v \cdot \left(\mathsf{neg}\left(t1\right)\right) + \color{blue}{0}}{u}}{u} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(v, \mathsf{neg}\left(t1\right), 0\right)}}{u}}{u} \]

      12. neg-sub0 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(v, \color{blue}{0 - t1}, 0\right)}{u}}{u} \]

      13. --lowering--.f64 95.1

          \[\leadsto \frac{\frac{\mathsf{fma}\left(v, \color{blue}{0 - t1}, 0\right)}{u}}{u} \]

   9. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v, 0 - t1, 0\right)}{u}}{u}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \frac{\frac{\color{blue}{v \cdot \left(0 - t1\right)}}{u}}{u} \]

       2. sub0-neg N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sub0-neg N/A

          \[\leadsto \frac{\color{blue}{\left(0 - t1\right)} \cdot \frac{v}{u}}{u} \]

       7. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(0 - t1\right)} \cdot \frac{v}{u}}{u} \]

       8. /-lowering-/.f64 99.9

          \[\leadsto \frac{\left(0 - t1\right) \cdot \color{blue}{\frac{v}{u}}}{u} \]

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{\color{blue}{\left(0 - t1\right) \cdot \frac{v}{u}}}{u} \]

   if -3.80000000000000016e118 < u

   1. Initial program 69.7%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{0 - v}{\color{blue}{t1 + u}} \cdot \frac{t1}{t1 + u} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      13. +-lowering-+.f64 98.4

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

   4. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{0 - v}{t1 + u} \cdot \frac{t1}{t1 + u}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{0 - v}{t1 + u}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{t1}}} \cdot \frac{0 - v}{t1 + u} \]

      3. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(0 - v\right)}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1 \cdot \left(0 - v\right)\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 - v\right)}\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)} \]

      6. flip3-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)} \]

      7. flip3-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 - v\right)}\right)}{\mathsf{neg}\left(\frac{t1 + u}{t1} \cdot \left(t1 + u\right)\right)} \]

      8. sub0-neg N/A

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

      9. remove-double-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. +-lowering-+.f64 96.3

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in t1 around inf

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

   9. Simplified 96.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118}:\\ \;\;\;\;0 - \frac{t1 \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \mathbf{if}\;u \leq -4.3 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{-112}:\\ \;\;\;\;0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- 0.0 (* t1 (/ v (* u (+ t1 u)))))))
   (if (<= u -4.3e-36)
     t_1
     (if (<= u 1.7e-112) (- 0.0 (/ v (fma u 2.0 t1))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 0.0 - (t1 * (v / (u * (t1 + u))));
	double tmp;
	if (u <= -4.3e-36) {
		tmp = t_1;
	} else if (u <= 1.7e-112) {
		tmp = 0.0 - (v / fma(u, 2.0, t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(0.0 - Float64(t1 * Float64(v / Float64(u * Float64(t1 + u)))))
	tmp = 0.0
	if (u <= -4.3e-36)
		tmp = t_1;
	elseif (u <= 1.7e-112)
		tmp = Float64(0.0 - Float64(v / fma(u, 2.0, t1)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(0.0 - N[(t1 * N[(v / N[(u * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.3e-36], t$95$1, If[LessEqual[u, 1.7e-112], N[(0.0 - N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -4.3000000000000002e-36 or 1.6999999999999999e-112 < u

   1. Initial program 80.2%

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

   3. Taylor expanded in t1 around 0

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

   5. Simplified 73.8%

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 77.1

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

   7. Applied egg-rr 77.1%

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

   if -4.3000000000000002e-36 < u < 1.6999999999999999e-112

   1. Initial program 57.4%

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

   3. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 51.2

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

   5. Simplified 51.2%

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

      1. times-frac N/A

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

      2. distribute-frac-neg N/A

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

      3. *-inverses N/A

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

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

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. accelerator-lowering-fma.f64 82.9

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{-36}:\\ \;\;\;\;0 - t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{-112}:\\ \;\;\;\;0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0 - t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -4.3 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-16}:\\ \;\;\;\;0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- 0.0 t1) (/ v (* u u)))))
   (if (<= u -4.3e-36)
     t_1
     (if (<= u 3e-16) (- 0.0 (/ v (fma u 2.0 t1))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (0.0 - t1) * (v / (u * u));
	double tmp;
	if (u <= -4.3e-36) {
		tmp = t_1;
	} else if (u <= 3e-16) {
		tmp = 0.0 - (v / fma(u, 2.0, t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(0.0 - t1) * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -4.3e-36)
		tmp = t_1;
	elseif (u <= 3e-16)
		tmp = Float64(0.0 - Float64(v / fma(u, 2.0, t1)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(0.0 - t1), $MachinePrecision] * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.3e-36], t$95$1, If[LessEqual[u, 3e-16], N[(0.0 - N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -4.3000000000000002e-36 or 2.99999999999999994e-16 < u

   1. Initial program 82.0%

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \cdot \frac{t1}{t1 + u} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{0 - v}{\color{blue}{t1 + u}} \cdot \frac{t1}{t1 + u} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \color{blue}{\frac{t1}{t1 + u}} \]

      13. +-lowering-+.f64 97.8

          \[\leadsto \frac{0 - v}{t1 + u} \cdot \frac{t1}{\color{blue}{t1 + u}} \]

   4. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{0 - v}{t1 + u} \cdot \frac{t1}{t1 + u}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-frac-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-lowering-*.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 92.8

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in t1 around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 79.9

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

   9. Simplified 79.9%

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

   if -4.3000000000000002e-36 < u < 2.99999999999999994e-16

   1. Initial program 59.3%

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

   3. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 47.7

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

   5. Simplified 47.7%

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

      1. times-frac N/A

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

      2. distribute-frac-neg N/A

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

      3. *-inverses N/A

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

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

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. accelerator-lowering-fma.f64 77.0

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

   7. Applied egg-rr 77.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{-36}:\\ \;\;\;\;\left(0 - t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-16}:\\ \;\;\;\;0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0 - t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+118) (* (/ v u) -0.5) (/ v (- 0.0 t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+118) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.8d+118)) then
        tmp = (v / u) * (-0.5d0)
    else
        tmp = v / (0.0d0 - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+118) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e+118:
		tmp = (v / u) * -0.5
	else:
		tmp = v / (0.0 - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+118)
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(v / Float64(0.0 - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e+118)
		tmp = (v / u) * -0.5;
	else
		tmp = v / (0.0 - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+118], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.80000000000000016e118

   1. Initial program 80.5%

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

   3. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 45.5

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

   5. Simplified 45.5%

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

   6. Taylor expanded in t1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 36.7

         \[\leadsto \color{blue}{\frac{v}{u}} \cdot -0.5 \]

   8. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   if -3.80000000000000016e118 < u

   1. Initial program 69.7%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]

      4. /-lowering-/.f64 54.3

         \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 54.3

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

   7. Applied egg-rr 54.3%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118}:\\ \;\;\;\;0 - \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+118) (- 0.0 (/ v u)) (/ v (- 0.0 t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+118) {
		tmp = 0.0 - (v / u);
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.8d+118)) then
        tmp = 0.0d0 - (v / u)
    else
        tmp = v / (0.0d0 - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+118) {
		tmp = 0.0 - (v / u);
	} else {
		tmp = v / (0.0 - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e+118:
		tmp = 0.0 - (v / u)
	else:
		tmp = v / (0.0 - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+118)
		tmp = Float64(0.0 - Float64(v / u));
	else
		tmp = Float64(v / Float64(0.0 - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e+118)
		tmp = 0.0 - (v / u);
	else
		tmp = v / (0.0 - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+118], N[(0.0 - N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.80000000000000016e118

   1. Initial program 80.5%

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

   3. Taylor expanded in t1 around 0

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

   5. Simplified 80.5%

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

   6. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{v}{u}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{v}{u}} \]

      4. /-lowering-/.f64 36.6

         \[\leadsto 0 - \color{blue}{\frac{v}{u}} \]

   8. Simplified 36.6%

      \[\leadsto \color{blue}{0 - \frac{v}{u}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 36.6

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

   10. Applied egg-rr 36.6%

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

   if -3.80000000000000016e118 < u

   1. Initial program 69.7%

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

   3. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]

      4. /-lowering-/.f64 54.3

         \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 54.3

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

   7. Applied egg-rr 54.3%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118}:\\ \;\;\;\;0 - \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{0 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (- 0.0 (/ v (fma u 2.0 t1))))

C

double code(double u, double v, double t1) {
	return 0.0 - (v / fma(u, 2.0, t1));
}

Julia

function code(u, v, t1)
	return Float64(0.0 - Float64(v / fma(u, 2.0, t1)))
end

Wolfram

code[u_, v_, t1_] := N[(0.0 - N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 39.9

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

5. Simplified 39.9%

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

   1. times-frac N/A

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

   2. distribute-frac-neg N/A

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

   3. *-inverses N/A

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

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

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. neg-lowering-neg.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. accelerator-lowering-fma.f64 54.4

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

7. Applied egg-rr 54.4%

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

8. Final simplification 54.4%

   \[\leadsto 0 - \frac{v}{\mathsf{fma}\left(u, 2, t1\right)} \]
9. Add Preprocessing

Alternative 10: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(0 - t1\right) - u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (- 0.0 t1) u)))

C

double code(double u, double v, double t1) {
	return v / ((0.0 - 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 / ((0.0d0 - t1) - u)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((0.0 - t1) - u);
}

Python

def code(u, v, t1):
	return v / ((0.0 - t1) - u)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(0.0 - t1) - u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((0.0 - t1) - u);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(0.0 - t1), $MachinePrecision] - u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{0 - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot v\right)}{t1 + u}}}{t1 + u} \]

   6. --lowering--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot v\right)}{t1 + u}}}{t1 + u} \]

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

      \[\leadsto \frac{0 - \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t1 \cdot v\right)\right)}\right)}{t1 + u}}{t1 + u} \]

   8. remove-double-neg N/A

      \[\leadsto \frac{0 - \frac{\color{blue}{t1 \cdot v}}{t1 + u}}{t1 + u} \]

   9. associate-/l* N/A

      \[\leadsto \frac{0 - \color{blue}{t1 \cdot \frac{v}{t1 + u}}}{t1 + u} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{0 - \color{blue}{t1 \cdot \frac{v}{t1 + u}}}{t1 + u} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \frac{0 - t1 \cdot \color{blue}{\frac{v}{t1 + u}}}{t1 + u} \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \frac{0 - t1 \cdot \frac{v}{\color{blue}{t1 + u}}}{t1 + u} \]

   13. +-lowering-+.f64 97.7

       \[\leadsto \frac{0 - t1 \cdot \frac{v}{t1 + u}}{\color{blue}{t1 + u}} \]

4. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\frac{0 - t1 \cdot \frac{v}{t1 + u}}{t1 + u}} \]

5. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \]

   3. --lowering--.f64 53.7

      \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \]

7. Simplified 53.7%

   \[\leadsto \frac{\color{blue}{0 - v}}{t1 + u} \]

8. Final simplification 53.7%

   \[\leadsto \frac{v}{\left(0 - t1\right) - u} \]
9. Add Preprocessing

Alternative 11: 53.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{0 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- 0.0 t1)))

C

double code(double u, double v, double t1) {
	return v / (0.0 - 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 / (0.0d0 - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (0.0 - t1);
}

Python

def code(u, v, t1):
	return v / (0.0 - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(0.0 - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (0.0 - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(0.0 - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

3. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]

   4. /-lowering-/.f64 47.1

      \[\leadsto 0 - \color{blue}{\frac{v}{t1}} \]

5. Simplified 47.1%

   \[\leadsto \color{blue}{0 - \frac{v}{t1}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. /-lowering-/.f64 47.1

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

7. Applied egg-rr 47.1%

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

8. Final simplification 47.1%

   \[\leadsto \frac{v}{0 - t1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))