Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.3%

Time: 10.6s

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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. *-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. times-frac N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 99.0

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

4. Applied egg-rr 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;-\frac{t1 \cdot \frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))))
        (t_2 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -5e+112)
     t_2
     (if (<= t1 -1.7e-150)
       t_1
       (if (<= t1 2.2e-163)
         (- (/ (* t1 (/ v u)) u))
         (if (<= t1 3.2e+72) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5e+112) {
		tmp = t_2;
	} else if (t1 <= -1.7e-150) {
		tmp = t_1;
	} else if (t1 <= 2.2e-163) {
		tmp = -((t1 * (v / u)) / u);
	} else if (t1 <= 3.2e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    t_2 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-5d+112)) then
        tmp = t_2
    else if (t1 <= (-1.7d-150)) then
        tmp = t_1
    else if (t1 <= 2.2d-163) then
        tmp = -((t1 * (v / u)) / u)
    else if (t1 <= 3.2d+72) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5e+112) {
		tmp = t_2;
	} else if (t1 <= -1.7e-150) {
		tmp = t_1;
	} else if (t1 <= 2.2e-163) {
		tmp = -((t1 * (v / u)) / u);
	} else if (t1 <= 3.2e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	t_2 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -5e+112:
		tmp = t_2
	elif t1 <= -1.7e-150:
		tmp = t_1
	elif t1 <= 2.2e-163:
		tmp = -((t1 * (v / u)) / u)
	elif t1 <= 3.2e+72:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	t_2 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -5e+112)
		tmp = t_2;
	elseif (t1 <= -1.7e-150)
		tmp = t_1;
	elseif (t1 <= 2.2e-163)
		tmp = Float64(-Float64(Float64(t1 * Float64(v / u)) / u));
	elseif (t1 <= 3.2e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	t_2 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -5e+112)
		tmp = t_2;
	elseif (t1 <= -1.7e-150)
		tmp = t_1;
	elseif (t1 <= 2.2e-163)
		tmp = -((t1 * (v / u)) / u);
	elseif (t1 <= 3.2e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5e+112], t$95$2, If[LessEqual[t1, -1.7e-150], t$95$1, If[LessEqual[t1, 2.2e-163], (-N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]), If[LessEqual[t1, 3.2e+72], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5e112 or 3.2000000000000001e72 < t1

   1. Initial program 46.2%

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

      1. *-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. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-frac-neg N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/l* N/A

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

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

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

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

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

      12. frac-2neg N/A

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

      13. remove-double-neg N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-frac-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 96.5

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 92.0

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

   9. Simplified 92.0%

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

   if -5e112 < t1 < -1.7e-150 or 2.20000000000000011e-163 < t1 < 3.2000000000000001e72

   1. Initial program 92.2%

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

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

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

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

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

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

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

      9. +-lowering-+.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   if -1.7e-150 < t1 < 2.20000000000000011e-163

   1. Initial program 77.6%

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

      1. *-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. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 97.0

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in t1 around 0

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

      1. /-lowering-/.f64 92.6

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

   7. Simplified 92.6%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 92.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-150}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;-\frac{t1 \cdot \frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+72}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+112}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -3.25 \cdot 10^{-180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5 \cdot 10^{-164}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))))
        (t_2 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -5.5e+112)
     t_2
     (if (<= t1 -3.25e-180)
       t_1
       (if (<= t1 5e-164)
         (* (/ v u) (/ t1 (- u)))
         (if (<= t1 3.2e+72) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5.5e+112) {
		tmp = t_2;
	} else if (t1 <= -3.25e-180) {
		tmp = t_1;
	} else if (t1 <= 5e-164) {
		tmp = (v / u) * (t1 / -u);
	} else if (t1 <= 3.2e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    t_2 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-5.5d+112)) then
        tmp = t_2
    else if (t1 <= (-3.25d-180)) then
        tmp = t_1
    else if (t1 <= 5d-164) then
        tmp = (v / u) * (t1 / -u)
    else if (t1 <= 3.2d+72) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5.5e+112) {
		tmp = t_2;
	} else if (t1 <= -3.25e-180) {
		tmp = t_1;
	} else if (t1 <= 5e-164) {
		tmp = (v / u) * (t1 / -u);
	} else if (t1 <= 3.2e+72) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	t_2 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -5.5e+112:
		tmp = t_2
	elif t1 <= -3.25e-180:
		tmp = t_1
	elif t1 <= 5e-164:
		tmp = (v / u) * (t1 / -u)
	elif t1 <= 3.2e+72:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	t_2 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -5.5e+112)
		tmp = t_2;
	elseif (t1 <= -3.25e-180)
		tmp = t_1;
	elseif (t1 <= 5e-164)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	elseif (t1 <= 3.2e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	t_2 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -5.5e+112)
		tmp = t_2;
	elseif (t1 <= -3.25e-180)
		tmp = t_1;
	elseif (t1 <= 5e-164)
		tmp = (v / u) * (t1 / -u);
	elseif (t1 <= 3.2e+72)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.5e+112], t$95$2, If[LessEqual[t1, -3.25e-180], t$95$1, If[LessEqual[t1, 5e-164], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.2e+72], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.50000000000000026e112 or 3.2000000000000001e72 < t1

   1. Initial program 46.2%

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

      1. *-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. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-frac-neg N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/l* N/A

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

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

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

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

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

      12. frac-2neg N/A

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

      13. remove-double-neg N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-frac-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 96.5

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 92.0

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

   9. Simplified 92.0%

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

   if -5.50000000000000026e112 < t1 < -3.25000000000000007e-180 or 4.99999999999999962e-164 < t1 < 3.2000000000000001e72

   1. Initial program 91.7%

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

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

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

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

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

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

      9. +-lowering-+.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   if -3.25000000000000007e-180 < t1 < 4.99999999999999962e-164

   1. Initial program 77.3%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.f64 N/A

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

      14. neg-lowering-neg.f64 78.8

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

   5. Simplified 78.8%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 90.6

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

   7. Applied egg-rr 90.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -3.25 \cdot 10^{-180}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 5 \cdot 10^{-164}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+72}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -2.15e-63)
     t_1
     (if (<= t1 2.9e-80) (* (/ v u) (/ t1 (- u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.15e-63) {
		tmp = t_1;
	} else if (t1 <= 2.9e-80) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-2.15d-63)) then
        tmp = t_1
    else if (t1 <= 2.9d-80) then
        tmp = (v / u) * (t1 / -u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.15e-63) {
		tmp = t_1;
	} else if (t1 <= 2.9e-80) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2.15e-63:
		tmp = t_1
	elif t1 <= 2.9e-80:
		tmp = (v / u) * (t1 / -u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -2.15e-63)
		tmp = t_1;
	elseif (t1 <= 2.9e-80)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2.15e-63)
		tmp = t_1;
	elseif (t1 <= 2.9e-80)
		tmp = (v / u) * (t1 / -u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.15e-63], t$95$1, If[LessEqual[t1, 2.9e-80], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.1499999999999999e-63 or 2.89999999999999998e-80 < t1

   1. Initial program 66.0%

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

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-frac-neg N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/l* N/A

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

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

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

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

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

      12. frac-2neg N/A

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

      13. remove-double-neg N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-frac-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 95.4

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

   6. Applied egg-rr 95.4%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 84.8

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

   9. Simplified 84.8%

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

   if -2.1499999999999999e-63 < t1 < 2.89999999999999998e-80

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

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

      14. neg-lowering-neg.f64 79.2

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

   5. Simplified 79.2%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 87.4

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

   7. Applied egg-rr 87.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 4.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 4.8e+210)
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
   (- (/ (* t1 (/ v u)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 4.8e+210) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = -((t1 * (v / u)) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 4.8d+210) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = -((t1 * (v / u)) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 4.8e+210) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = -((t1 * (v / u)) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 4.8e+210:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = -((t1 * (v / u)) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 4.8e+210)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(-Float64(Float64(t1 * Float64(v / u)) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 4.8e+210)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = -((t1 * (v / u)) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 4.8e+210], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < 4.79999999999999977e210

   1. Initial program 72.1%

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

      1. *-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. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 98.9

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

   4. Applied egg-rr 98.9%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-frac-neg N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/l* N/A

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

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

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

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

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

      12. frac-2neg N/A

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

      13. remove-double-neg N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-frac-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 95.8

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

   6. Applied egg-rr 95.8%

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

   if 4.79999999999999977e210 < u

   1. Initial program 82.2%

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

      1. *-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. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 99.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 4.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-79}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -2.6e-63)
     t_1
     (if (<= t1 1.26e-79) (* t1 (/ v (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.6e-63) {
		tmp = t_1;
	} else if (t1 <= 1.26e-79) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-2.6d-63)) then
        tmp = t_1
    else if (t1 <= 1.26d-79) then
        tmp = t1 * (v / (u * -u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.6e-63) {
		tmp = t_1;
	} else if (t1 <= 1.26e-79) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2.6e-63:
		tmp = t_1
	elif t1 <= 1.26e-79:
		tmp = t1 * (v / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -2.6e-63)
		tmp = t_1;
	elseif (t1 <= 1.26e-79)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2.6e-63)
		tmp = t_1;
	elseif (t1 <= 1.26e-79)
		tmp = t1 * (v / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.6e-63], t$95$1, If[LessEqual[t1, 1.26e-79], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.6000000000000001e-63 or 1.25999999999999993e-79 < t1

   1. Initial program 66.0%

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

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg2 N/A

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

      3. distribute-frac-neg N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/l* N/A

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

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

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

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

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

      12. frac-2neg N/A

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

      13. remove-double-neg N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-frac-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 95.4

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

   6. Applied egg-rr 95.4%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 84.8

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

   9. Simplified 84.8%

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

   if -2.6000000000000001e-63 < t1 < 1.25999999999999993e-79

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

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

      14. neg-lowering-neg.f64 79.2

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

   5. Simplified 79.2%

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

Alternative 7: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

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

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

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

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

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

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

   12. +-lowering-+.f64 98.0

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

4. Applied egg-rr 98.0%

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

5. Final simplification 98.0%

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

Alternative 8: 57.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -8.4 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 9.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v u))))
   (if (<= u -8.4e+193) t_1 (if (<= u 9.6e+208) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -8.4e+193) {
		tmp = t_1;
	} else if (u <= 9.6e+208) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(v / u)
    if (u <= (-8.4d+193)) then
        tmp = t_1
    else if (u <= 9.6d+208) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -8.4e+193) {
		tmp = t_1;
	} else if (u <= 9.6e+208) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / u)
	tmp = 0
	if u <= -8.4e+193:
		tmp = t_1
	elif u <= 9.6e+208:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / u))
	tmp = 0.0
	if (u <= -8.4e+193)
		tmp = t_1;
	elseif (u <= 9.6e+208)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / u);
	tmp = 0.0;
	if (u <= -8.4e+193)
		tmp = t_1;
	elseif (u <= 9.6e+208)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / u), $MachinePrecision])}, If[LessEqual[u, -8.4e+193], t$95$1, If[LessEqual[u, 9.6e+208], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -8.4e193 or 9.59999999999999946e208 < u

   1. Initial program 82.5%

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

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

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

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

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

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

      9. +-lowering-+.f64 82.9

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

   4. Applied egg-rr 82.9%

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

   5. Taylor expanded in t1 around 0

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

   7. Simplified 82.9%

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

   8. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 36.7

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

   10. Simplified 36.7%

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

   if -8.4e193 < u < 9.59999999999999946e208

   1. Initial program 71.2%

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

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. neg-lowering-neg.f64 62.4

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

   5. Simplified 62.4%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.4 \cdot 10^{+193}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

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

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - 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 / ((u * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

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

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. *-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. times-frac N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 99.0

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

4. Applied egg-rr 99.0%

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

   1. associate-/l* N/A

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

   2. distribute-frac-neg2 N/A

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

   3. distribute-frac-neg N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

   9. associate-/l* N/A

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

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

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

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

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

   12. frac-2neg N/A

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

   13. remove-double-neg N/A

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

   14. distribute-neg-in N/A

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

   15. unsub-neg N/A

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

   16. div-sub N/A

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

   17. distribute-frac-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. --lowering--.f64 N/A

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

   21. /-lowering-/.f64 94.4

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

6. Applied egg-rr 94.4%

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

7. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 59.1

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

9. Simplified 59.1%

   \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
10. Add Preprocessing

Alternative 10: 61.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. *-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. times-frac N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 99.0

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 58.7%

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

8. Final simplification 58.7%

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

Alternative 11: 16.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

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

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

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

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

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

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

   9. +-lowering-+.f64 76.6

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

4. Applied egg-rr 76.6%

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

5. Taylor expanded in t1 around 0

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

7. Simplified 45.4%

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

8. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 13.9

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

10. Simplified 13.9%

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

11. Final simplification 13.9%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))