Rosa's DopplerBench

Percentage Accurate: 72.3% → 94.8%

Time: 9.0s

Alternatives: 8

Speedup: 1.0×

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: 72.3% 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: 94.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in t1 around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 55.4

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

5. Simplified 55.4%

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

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 88.6

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

8. Simplified 88.6%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-/.f64 95.9

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

11. Simplified 95.9%

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

12. Final simplification 95.9%

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

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{v \cdot \left(-t1\right)}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (fma u 2.0 t1)))))
   (if (<= t1 -4e+138)
     t_1
     (if (<= t1 7.8e+123) (/ (* v (- t1)) (* (+ u t1) (+ u t1))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / fma(u, 2.0, t1));
	double tmp;
	if (t1 <= -4e+138) {
		tmp = t_1;
	} else if (t1 <= 7.8e+123) {
		tmp = (v * -t1) / ((u + t1) * (u + t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / fma(u, 2.0, t1)))
	tmp = 0.0
	if (t1 <= -4e+138)
		tmp = t_1;
	elseif (t1 <= 7.8e+123)
		tmp = Float64(Float64(v * Float64(-t1)) / Float64(Float64(u + t1) * Float64(u + t1)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -4e+138], t$95$1, If[LessEqual[t1, 7.8e+123], N[(N[(v * (-t1)), $MachinePrecision] / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.0000000000000001e138 or 7.79999999999999986e123 < t1

   1. Initial program 53.4%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 95.9

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

   8. Simplified 95.9%

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

   9. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 92.0

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

   11. Simplified 92.0%

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

   if -4.0000000000000001e138 < t1 < 7.79999999999999986e123

   1. Initial program 85.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+138}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{v \cdot \left(-t1\right)}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{\frac{u \cdot \left(-u\right)}{t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (fma u 2.0 t1)))))
   (if (<= t1 -5.8e-55)
     t_1
     (if (<= t1 3.9e-109) (/ v (/ (* u (- u)) t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / fma(u, 2.0, t1));
	double tmp;
	if (t1 <= -5.8e-55) {
		tmp = t_1;
	} else if (t1 <= 3.9e-109) {
		tmp = v / ((u * -u) / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / fma(u, 2.0, t1)))
	tmp = 0.0
	if (t1 <= -5.8e-55)
		tmp = t_1;
	elseif (t1 <= 3.9e-109)
		tmp = Float64(v / Float64(Float64(u * Float64(-u)) / t1));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -5.8e-55], t$95$1, If[LessEqual[t1, 3.9e-109], N[(v / N[(N[(u * (-u)), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.8e-55 or 3.90000000000000023e-109 < t1

   1. Initial program 72.2%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 70.9

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

   5. Simplified 70.9%

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

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 94.5

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

   8. Simplified 94.5%

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

   9. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 83.2

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

   11. Simplified 83.2%

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

   if -5.8e-55 < t1 < 3.90000000000000023e-109

   1. Initial program 82.6%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 34.8

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

   5. Simplified 34.8%

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

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 80.7

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

   8. Simplified 80.7%

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

   9. Taylor expanded in t1 around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 82.2

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

   11. Simplified 82.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-55}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{v}{\frac{u \cdot \left(-u\right)}{t1}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{if}\;u \leq -8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v (* u (- u))))))
   (if (<= u -8e-8) t_1 (if (<= u 4.7e+45) (- (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * -u));
	double tmp;
	if (u <= -8e-8) {
		tmp = t_1;
	} else if (u <= 4.7e+45) {
		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 <= (-8d-8)) then
        tmp = t_1
    else if (u <= 4.7d+45) 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 <= -8e-8) {
		tmp = t_1;
	} else if (u <= 4.7e+45) {
		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 <= -8e-8:
		tmp = t_1
	elif u <= 4.7e+45:
		tmp = -(v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / Float64(u * Float64(-u))))
	tmp = 0.0
	if (u <= -8e-8)
		tmp = t_1;
	elseif (u <= 4.7e+45)
		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 <= -8e-8)
		tmp = t_1;
	elseif (u <= 4.7e+45)
		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, -8e-8], t$95$1, If[LessEqual[u, 4.7e+45], (-N[(v / t1), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -8.0000000000000002e-8 or 4.70000000000000002e45 < u

   1. Initial program 86.2%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 84.4

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

   5. Simplified 84.4%

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

   if -8.0000000000000002e-8 < u < 4.70000000000000002e45

   1. Initial program 68.4%

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

   3. Taylor expanded in t1 around inf

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

      1. 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 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{-8}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in t1 around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 55.4

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

5. Simplified 55.4%

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

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 88.6

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

8. Simplified 88.6%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-/.f64 95.9

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

11. Simplified 95.9%

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

12. Taylor expanded in u around inf

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

    1. lower-/.f64 94.5

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

14. Simplified 94.5%

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

15. Final simplification 94.5%

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

Alternative 6: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v \cdot -0.5}{u}\\ \mathbf{if}\;u \leq -3.8 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* v -0.5) u)))
   (if (<= u -3.8e+79) t_1 (if (<= u 3.1e+105) (- (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v * -0.5) / u;
	double tmp;
	if (u <= -3.8e+79) {
		tmp = t_1;
	} else if (u <= 3.1e+105) {
		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 * (-0.5d0)) / u
    if (u <= (-3.8d+79)) then
        tmp = t_1
    else if (u <= 3.1d+105) 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 * -0.5) / u;
	double tmp;
	if (u <= -3.8e+79) {
		tmp = t_1;
	} else if (u <= 3.1e+105) {
		tmp = -(v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v * -0.5) / u
	tmp = 0
	if u <= -3.8e+79:
		tmp = t_1
	elif u <= 3.1e+105:
		tmp = -(v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v * -0.5) / u)
	tmp = 0.0
	if (u <= -3.8e+79)
		tmp = t_1;
	elseif (u <= 3.1e+105)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v * -0.5) / u;
	tmp = 0.0;
	if (u <= -3.8e+79)
		tmp = t_1;
	elseif (u <= 3.1e+105)
		tmp = -(v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v * -0.5), $MachinePrecision] / u), $MachinePrecision]}, If[LessEqual[u, -3.8e+79], t$95$1, If[LessEqual[u, 3.1e+105], (-N[(v / t1), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.8000000000000002e79 or 3.10000000000000004e105 < u

   1. Initial program 84.7%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 53.6

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

   5. Simplified 53.6%

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

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 90.3

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

   8. Simplified 90.3%

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

   9. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 50.5

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

   11. Simplified 50.5%

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

   12. Taylor expanded in u around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 45.6

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

   14. Simplified 45.6%

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

   if -3.8000000000000002e79 < u < 3.10000000000000004e105

   1. Initial program 72.5%

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

   3. Taylor expanded in t1 around 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 70.5

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

   5. Simplified 70.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in t1 around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{{t1}^{2} \cdot \left(1 + \left(2 \cdot \frac{u}{t1} + \frac{{u}^{2}}{{t1}^{2}}\right)\right)}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 55.4

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

5. Simplified 55.4%

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

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1 \cdot \left(1 + \frac{u \cdot \left(2 + \frac{u}{t1}\right)}{t1}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 88.6

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

8. Simplified 88.6%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 63.1

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

11. Simplified 63.1%

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

Alternative 8: 53.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in t1 around inf

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

   1. 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 53.9

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

5. Simplified 53.9%

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

6. Final simplification 53.9%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))