Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.0%

Time: 17.3s

Alternatives: 13

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: 72.9% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

Alternative 2: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -5.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 5.4 \cdot 10^{+134}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -5.2e+111)
     (/ v (- u t1))
     (if (<= t1 5.4e+134) (* t1 (/ (/ v (+ t1 u)) t_1)) (/ v t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -5.2e+111) {
		tmp = v / (u - t1);
	} else if (t1 <= 5.4e+134) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = v / 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 = -u - t1
    if (t1 <= (-5.2d+111)) then
        tmp = v / (u - t1)
    else if (t1 <= 5.4d+134) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -5.2e+111) {
		tmp = v / (u - t1);
	} else if (t1 <= 5.4e+134) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -5.2e+111:
		tmp = v / (u - t1)
	elif t1 <= 5.4e+134:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -5.2e+111)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 5.4e+134)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -5.2e+111)
		tmp = v / (u - t1);
	elseif (t1 <= 5.4e+134)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -5.2e+111], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.4e+134], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(v / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.1999999999999997e111

   1. Initial program 38.6%

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

      1. associate-*l/ 40.2%

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

      2. *-commutative 40.2%

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

   3. Simplified 40.2%

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

      1. associate-*r/ 38.6%

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

      2. *-commutative 38.6%

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

      3. times-frac 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 99.4%

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

      10. sqrt-unprod 19.0%

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

      11. sqr-neg 19.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 25.5%

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

      14. sub-neg 25.5%

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

      15. +-commutative 25.5%

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

      16. add-sqr-sqrt 25.5%

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

      17. sqrt-unprod 26.4%

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

      18. sqr-neg 26.4%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 0.0%

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

      22. sqr-neg 0.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 80.9%

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

      1. mul-1-neg 80.9%

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

   9. Simplified 80.9%

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

       1. add-sqr-sqrt 43.4%

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

       2. sqrt-unprod 72.9%

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

       3. sqr-neg 72.9%

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

       4. sqrt-unprod 37.3%

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

       5. add-sqr-sqrt 81.4%

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

       6. sub-neg 81.4%

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

   11. Applied egg-rr 81.4%

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

   if -5.1999999999999997e111 < t1 < 5.4e134

   1. Initial program 77.8%

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

      1. associate-/l* 81.7%

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

   3. Simplified 81.7%

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

      1. associate-/r* 89.5%

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

      2. div-inv 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. associate-*r/ 89.5%

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

      2. *-rgt-identity 89.5%

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

   8. Simplified 89.5%

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

   if 5.4e134 < t1

   1. Initial program 27.5%

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

      1. associate-*l/ 29.4%

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

      2. *-commutative 29.4%

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

   3. Simplified 29.4%

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

      1. associate-*r/ 27.5%

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

      2. *-commutative 27.5%

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

      3. times-frac 100.0%

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

      4. frac-2neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 3.3%

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

      11. sqr-neg 3.3%

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

      12. sqrt-unprod 20.7%

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

      13. add-sqr-sqrt 20.7%

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

      14. sub-neg 20.7%

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

      15. +-commutative 20.7%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 29.4%

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

      18. sqr-neg 29.4%

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

      19. sqrt-unprod 95.5%

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

      20. add-sqr-sqrt 44.9%

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

      21. sqrt-unprod 88.1%

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

      22. sqr-neg 88.1%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 96.4%

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

      1. mul-1-neg 96.4%

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

   9. Simplified 96.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 5.4 \cdot 10^{+134}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -1.12 \cdot 10^{+108}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{+134}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -1.12e+108)
     (/ v (- u t1))
     (if (<= t1 8.2e+134) (* v (/ t1 (* (+ t1 u) t_1))) (/ v t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -1.12e+108) {
		tmp = v / (u - t1);
	} else if (t1 <= 8.2e+134) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v / 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 = -u - t1
    if (t1 <= (-1.12d+108)) then
        tmp = v / (u - t1)
    else if (t1 <= 8.2d+134) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -1.12e+108) {
		tmp = v / (u - t1);
	} else if (t1 <= 8.2e+134) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -1.12e+108:
		tmp = v / (u - t1)
	elif t1 <= 8.2e+134:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -1.12e+108)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 8.2e+134)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -1.12e+108)
		tmp = v / (u - t1);
	elseif (t1 <= 8.2e+134)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -1.12e+108], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 8.2e+134], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.11999999999999994e108

   1. Initial program 38.4%

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

      1. associate-*l/ 42.3%

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

      2. *-commutative 42.3%

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

   3. Simplified 42.3%

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

      1. associate-*r/ 38.4%

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

      2. *-commutative 38.4%

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

      3. times-frac 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 99.4%

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

      10. sqrt-unprod 25.3%

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

      11. sqr-neg 25.3%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 23.6%

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

      14. sub-neg 23.6%

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

      15. +-commutative 23.6%

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

      16. add-sqr-sqrt 23.6%

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

      17. sqrt-unprod 24.5%

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

      18. sqr-neg 24.5%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 0.0%

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

      22. sqr-neg 0.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 79.9%

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

      1. mul-1-neg 79.9%

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

   9. Simplified 79.9%

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

       1. add-sqr-sqrt 40.0%

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

       2. sqrt-unprod 72.5%

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

       3. sqr-neg 72.5%

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

       4. sqrt-unprod 39.6%

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

       5. add-sqr-sqrt 80.4%

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

       6. sub-neg 80.4%

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

   11. Applied egg-rr 80.4%

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

   if -1.11999999999999994e108 < t1 < 8.2000000000000007e134

   1. Initial program 78.5%

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

      1. associate-*l/ 85.7%

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

      2. *-commutative 85.7%

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

   3. Simplified 85.7%

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

   if 8.2000000000000007e134 < t1

   1. Initial program 27.5%

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

      1. associate-*l/ 29.4%

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

      2. *-commutative 29.4%

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

   3. Simplified 29.4%

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

      1. associate-*r/ 27.5%

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

      2. *-commutative 27.5%

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

      3. times-frac 100.0%

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

      4. frac-2neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 3.3%

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

      11. sqr-neg 3.3%

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

      12. sqrt-unprod 20.7%

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

      13. add-sqr-sqrt 20.7%

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

      14. sub-neg 20.7%

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

      15. +-commutative 20.7%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 29.4%

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

      18. sqr-neg 29.4%

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

      19. sqrt-unprod 95.5%

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

      20. add-sqr-sqrt 44.9%

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

      21. sqrt-unprod 88.1%

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

      22. sqr-neg 88.1%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 96.4%

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

      1. mul-1-neg 96.4%

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

   9. Simplified 96.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{+108}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{+134}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{-6} \lor \neg \left(u \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{\frac{t1 + u}{t1}}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.25e-6) (not (<= u 3.6e-6)))
   (/ (* t1 (/ v u)) (- u))
   (/ (/ (- v) (/ (+ t1 u) t1)) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.25e-6) || !(u <= 3.6e-6)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = (-v / ((t1 + u) / t1)) / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.25d-6)) .or. (.not. (u <= 3.6d-6))) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = (-v / ((t1 + u) / t1)) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.25e-6) || !(u <= 3.6e-6)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = (-v / ((t1 + u) / t1)) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.25e-6) or not (u <= 3.6e-6):
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = (-v / ((t1 + u) / t1)) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.25e-6) || !(u <= 3.6e-6))
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(Float64(Float64(-v) / Float64(Float64(t1 + u) / t1)) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.25e-6) || ~((u <= 3.6e-6)))
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = (-v / ((t1 + u) / t1)) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.25e-6], N[Not[LessEqual[u, 3.6e-6]], $MachinePrecision]], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[(N[((-v) / N[(N[(t1 + u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.2500000000000001e-6 or 3.59999999999999984e-6 < u

   1. Initial program 71.5%

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

      1. associate-*l/ 68.0%

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

      2. *-commutative 68.0%

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

   3. Simplified 68.0%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

      3. times-frac 97.7%

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

      4. frac-2neg 97.7%

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

      5. +-commutative 97.7%

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

      6. distribute-neg-in 97.7%

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

      7. sub-neg 97.7%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 44.4%

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

      10. sqrt-unprod 59.1%

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

      11. sqr-neg 59.1%

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

      12. sqrt-unprod 27.2%

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

      13. add-sqr-sqrt 48.2%

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

      14. sub-neg 48.2%

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

      15. +-commutative 48.2%

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

      16. add-sqr-sqrt 21.0%

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

      17. sqrt-unprod 49.5%

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

      18. sqr-neg 49.5%

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

      19. sqrt-unprod 37.0%

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

      20. add-sqr-sqrt 19.0%

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

      21. sqrt-unprod 40.5%

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

      22. sqr-neg 40.5%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in t1 around 0 79.3%

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

   8. Taylor expanded in t1 around 0 78.6%

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

   9. Taylor expanded in t1 around 0 76.5%

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

       1. mul-1-neg 76.5%

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

       2. associate-*r/ 81.6%

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

       3. distribute-rgt-neg-out 81.6%

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

       4. distribute-neg-frac2 81.6%

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

   11. Simplified 81.6%

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

   if -1.2500000000000001e-6 < u < 3.59999999999999984e-6

   1. Initial program 54.5%

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

      1. associate-*l/ 68.1%

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

      2. *-commutative 68.1%

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

   3. Simplified 68.1%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. times-frac 97.9%

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

      4. frac-2neg 97.9%

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

      5. +-commutative 97.9%

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

      6. distribute-neg-in 97.9%

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

      7. sub-neg 97.9%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 43.1%

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

      10. sqrt-unprod 26.9%

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

      11. sqr-neg 26.9%

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

      12. sqrt-unprod 7.9%

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

      13. add-sqr-sqrt 13.2%

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

      14. sub-neg 13.2%

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

      15. +-commutative 13.2%

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

      16. add-sqr-sqrt 5.4%

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

      17. sqrt-unprod 29.0%

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

      18. sqr-neg 29.0%

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

      19. sqrt-unprod 42.6%

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

      20. add-sqr-sqrt 23.7%

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

      21. sqrt-unprod 48.0%

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

      22. sqr-neg 48.0%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in v around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. associate-*r/ 95.6%

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

      3. *-commutative 95.6%

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

      4. associate-/r/ 97.6%

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

      5. distribute-neg-frac2 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in t1 around inf 80.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{-6} \lor \neg \left(u \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{\frac{t1 + u}{t1}}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-6} \lor \neg \left(u \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7e-6) (not (<= u 5e-8)))
   (/ (* t1 (/ v (- u))) u)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e-6) || !(u <= 5e-8)) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.7d-6)) .or. (.not. (u <= 5d-8))) then
        tmp = (t1 * (v / -u)) / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e-6) || !(u <= 5e-8)) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7e-6) or not (u <= 5e-8):
		tmp = (t1 * (v / -u)) / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7e-6) || !(u <= 5e-8))
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.7e-6) || ~((u <= 5e-8)))
		tmp = (t1 * (v / -u)) / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7e-6], N[Not[LessEqual[u, 5e-8]], $MachinePrecision]], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.70000000000000003e-6 or 4.9999999999999998e-8 < u

   1. Initial program 71.5%

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

      1. associate-*l/ 68.0%

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

      2. *-commutative 68.0%

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

   3. Simplified 68.0%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

      3. times-frac 97.7%

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

      4. frac-2neg 97.7%

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

      5. +-commutative 97.7%

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

      6. distribute-neg-in 97.7%

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

      7. sub-neg 97.7%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 44.4%

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

      10. sqrt-unprod 59.1%

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

      11. sqr-neg 59.1%

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

      12. sqrt-unprod 27.2%

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

      13. add-sqr-sqrt 48.2%

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

      14. sub-neg 48.2%

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

      15. +-commutative 48.2%

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

      16. add-sqr-sqrt 21.0%

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

      17. sqrt-unprod 49.5%

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

      18. sqr-neg 49.5%

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

      19. sqrt-unprod 37.0%

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

      20. add-sqr-sqrt 19.0%

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

      21. sqrt-unprod 40.5%

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

      22. sqr-neg 40.5%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in t1 around 0 79.3%

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

   8. Taylor expanded in t1 around 0 78.6%

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

   9. Taylor expanded in t1 around 0 76.5%

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

       1. mul-1-neg 76.5%

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

       2. associate-*r/ 81.6%

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

       3. distribute-rgt-neg-out 81.6%

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

       4. distribute-neg-frac2 81.6%

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

   11. Simplified 81.6%

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

   if -1.70000000000000003e-6 < u < 4.9999999999999998e-8

   1. Initial program 54.5%

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

      1. associate-*l/ 68.1%

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

      2. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in t1 around inf 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-6} \lor \neg \left(u \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{-7} \lor \neg \left(u \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e-7) (not (<= u 5.8e-6)))
   (* (/ t1 (- u)) (/ v u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e-7) || !(u <= 5.8e-6)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.4d-7)) .or. (.not. (u <= 5.8d-6))) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e-7) || !(u <= 5.8e-6)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e-7) or not (u <= 5.8e-6):
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e-7) || !(u <= 5.8e-6))
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e-7) || ~((u <= 5.8e-6)))
		tmp = (t1 / -u) * (v / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e-7], N[Not[LessEqual[u, 5.8e-6]], $MachinePrecision]], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.39999999999999974e-7 or 5.8000000000000004e-6 < u

   1. Initial program 71.5%

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

      1. times-frac 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. unsub-neg 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t1 around 0 79.4%

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

   6. Taylor expanded in t1 around 0 79.5%

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

      1. associate-*r/ 79.5%

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

      2. mul-1-neg 79.5%

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

   8. Simplified 79.5%

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

   if -3.39999999999999974e-7 < u < 5.8000000000000004e-6

   1. Initial program 54.5%

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

      1. associate-*l/ 68.1%

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

      2. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in t1 around inf 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{-7} \lor \neg \left(u \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+166} \lor \neg \left(u \leq 4.8 \cdot 10^{+183}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3e+166) (not (<= u 4.8e+183)))
   (* v (/ (/ t1 u) u))
   (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e+166) || !(u <= 4.8e+183)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3d+166)) .or. (.not. (u <= 4.8d+183))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e+166) || !(u <= 4.8e+183)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3e+166) or not (u <= 4.8e+183):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3e+166) || !(u <= 4.8e+183))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3e+166) || ~((u <= 4.8e+183)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3e+166], N[Not[LessEqual[u, 4.8e+183]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.99999999999999998e166 or 4.8000000000000003e183 < u

   1. Initial program 70.7%

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

      1. associate-*l/ 71.5%

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

      2. *-commutative 71.5%

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

   3. Simplified 71.5%

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

      1. associate-*r/ 70.7%

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

      2. *-commutative 70.7%

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

      3. times-frac 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 47.9%

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

      10. sqrt-unprod 70.6%

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

      11. sqr-neg 70.6%

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

      12. sqrt-unprod 40.3%

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

      13. add-sqr-sqrt 71.0%

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

      14. sub-neg 71.0%

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

      15. +-commutative 71.0%

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

      16. add-sqr-sqrt 30.7%

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

      17. sqrt-unprod 71.4%

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

      18. sqr-neg 71.4%

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

      19. sqrt-unprod 44.2%

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

      20. add-sqr-sqrt 30.1%

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

      21. sqrt-unprod 40.5%

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

      22. sqr-neg 40.5%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around 0 92.0%

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

   8. Taylor expanded in t1 around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-/l* 77.1%

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

      3. add-sqr-sqrt 40.5%

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

      4. sqrt-unprod 66.4%

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

      5. sqr-neg 66.4%

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

      6. sqrt-unprod 32.6%

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

      7. add-sqr-sqrt 71.2%

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

   10. Applied egg-rr 71.2%

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

   if -2.99999999999999998e166 < u < 4.8000000000000003e183

   1. Initial program 60.4%

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

      1. associate-*l/ 67.2%

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

      2. *-commutative 67.2%

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

   3. Simplified 67.2%

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

      1. associate-*r/ 60.4%

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

      2. *-commutative 60.4%

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

      3. times-frac 97.3%

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

      4. frac-2neg 97.3%

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

      5. +-commutative 97.3%

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

      6. distribute-neg-in 97.3%

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

      7. sub-neg 97.3%

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

      8. associate-*r/ 97.0%

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

      9. add-sqr-sqrt 42.7%

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

      10. sqrt-unprod 34.9%

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

      11. sqr-neg 34.9%

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

      12. sqrt-unprod 11.1%

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

      13. add-sqr-sqrt 19.4%

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

      14. sub-neg 19.4%

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

      15. +-commutative 19.4%

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

      16. add-sqr-sqrt 8.2%

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

      17. sqrt-unprod 30.6%

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

      18. sqr-neg 30.6%

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

      19. sqrt-unprod 39.0%

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

      20. add-sqr-sqrt 19.5%

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

      21. sqrt-unprod 45.5%

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

      22. sqr-neg 45.5%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in t1 around inf 65.3%

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

      1. mul-1-neg 65.3%

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

   9. Simplified 65.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+166} \lor \neg \left(u \leq 4.8 \cdot 10^{+183}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+101} \lor \neg \left(t1 \leq 1.8 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5e+101) (not (<= t1 1.8e+88))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e+101) || !(t1 <= 1.8e+88)) {
		tmp = v / t1;
	} else {
		tmp = v / 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 ((t1 <= (-5d+101)) .or. (.not. (t1 <= 1.8d+88))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e+101) || !(t1 <= 1.8e+88)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5e+101) or not (t1 <= 1.8e+88):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5e+101) || !(t1 <= 1.8e+88))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5e+101) || ~((t1 <= 1.8e+88)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5e+101], N[Not[LessEqual[t1, 1.8e+88]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.99999999999999989e101 or 1.8000000000000001e88 < t1

   1. Initial program 36.1%

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

      1. associate-*l/ 39.5%

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

      2. *-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in t1 around inf 85.7%

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

      1. associate-*r/ 85.7%

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

      2. neg-mul-1 85.7%

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

   7. Simplified 85.7%

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

      1. add-sqr-sqrt 56.6%

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

      2. sqrt-unprod 50.5%

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

      3. sqr-neg 50.5%

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

      4. sqrt-unprod 8.1%

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

      5. add-sqr-sqrt 22.0%

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

      6. div-inv 22.0%

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

   9. Applied egg-rr 22.0%

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

       1. associate-*r/ 22.0%

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

       2. *-rgt-identity 22.0%

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

   11. Simplified 22.0%

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

   if -4.99999999999999989e101 < t1 < 1.8000000000000001e88

   1. Initial program 79.0%

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

      1. times-frac 96.4%

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

      2. distribute-frac-neg 96.4%

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

      3. distribute-neg-frac2 96.4%

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

      4. +-commutative 96.4%

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

      5. distribute-neg-in 96.4%

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

      6. unsub-neg 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t1 around 0 67.6%

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

   6. Taylor expanded in t1 around inf 18.7%

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

      1. associate-*r/ 18.7%

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

      2. mul-1-neg 18.7%

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

   8. Simplified 18.7%

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

      1. add-sqr-sqrt 10.9%

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

      2. sqrt-unprod 19.1%

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

      3. sqr-neg 19.1%

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

      4. sqrt-unprod 6.6%

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

      5. add-sqr-sqrt 16.2%

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

      6. div-inv 16.2%

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

   10. Applied egg-rr 16.2%

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

       1. associate-*r/ 16.2%

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

       2. *-rgt-identity 16.2%

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

   12. Simplified 16.2%

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

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+101} \lor \neg \left(t1 \leq 1.8 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2e+166) (/ 1.0 (/ u v)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2e+166) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2d+166)) then
        tmp = 1.0d0 / (u / v)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2e+166) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2e+166:
		tmp = 1.0 / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2e+166)
		tmp = Float64(1.0 / Float64(u / v));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2e+166)
		tmp = 1.0 / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2e+166], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.99999999999999988e166

   1. Initial program 78.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 92.8%

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

   6. Taylor expanded in t1 around inf 38.1%

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

      1. associate-*r/ 38.1%

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

      2. mul-1-neg 38.1%

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

   8. Simplified 38.1%

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

      1. clear-num 40.2%

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

      2. inv-pow 40.2%

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

      3. add-sqr-sqrt 23.5%

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

      4. sqrt-unprod 39.5%

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

      5. sqr-neg 39.5%

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

      6. sqrt-unprod 16.4%

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

      7. add-sqr-sqrt 39.8%

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

   10. Applied egg-rr 39.8%

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

       1. unpow-1 39.8%

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

   12. Simplified 39.8%

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

   if -1.99999999999999988e166 < u

   1. Initial program 60.5%

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

      1. associate-*l/ 66.8%

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

      2. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in t1 around inf 59.5%

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

      1. associate-*r/ 59.5%

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

      2. neg-mul-1 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.3e+165) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e+165) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.3d+165)) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e+165) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.3e+165:
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.3e+165)
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.3e+165)
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.3e+165], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.30000000000000016e165

   1. Initial program 78.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 92.8%

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

   6. Taylor expanded in t1 around inf 38.1%

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

      1. associate-*r/ 38.1%

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

      2. mul-1-neg 38.1%

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

   8. Simplified 38.1%

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

   if -2.30000000000000016e165 < u

   1. Initial program 60.5%

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

      1. associate-*l/ 66.8%

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

      2. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in t1 around inf 59.5%

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

      1. associate-*r/ 59.5%

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

      2. neg-mul-1 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.85e+165) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.85e+165) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.85d+165)) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.85e+165) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.85e+165:
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.85e+165)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.85e+165)
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.85e+165], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.85000000000000003e165

   1. Initial program 78.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 92.8%

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

   6. Taylor expanded in t1 around inf 38.1%

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

      1. associate-*r/ 38.1%

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

      2. mul-1-neg 38.1%

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

   8. Simplified 38.1%

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

      1. add-sqr-sqrt 23.5%

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

      2. sqrt-unprod 37.4%

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

      3. sqr-neg 37.4%

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

      4. sqrt-unprod 14.3%

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

      5. add-sqr-sqrt 37.8%

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

      6. div-inv 37.8%

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

   10. Applied egg-rr 37.8%

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

       1. associate-*r/ 37.8%

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

       2. *-rgt-identity 37.8%

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

   12. Simplified 37.8%

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

   if -1.85000000000000003e165 < u

   1. Initial program 60.5%

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

      1. associate-*l/ 66.8%

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

      2. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in t1 around inf 59.5%

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

      1. associate-*r/ 59.5%

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

      2. neg-mul-1 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. associate-*l/ 68.0%

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

   2. *-commutative 68.0%

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

3. Simplified 68.0%

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

   1. associate-*r/ 62.4%

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

   2. *-commutative 62.4%

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

   3. times-frac 97.8%

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

   4. frac-2neg 97.8%

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

   5. +-commutative 97.8%

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

   6. distribute-neg-in 97.8%

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

   7. sub-neg 97.8%

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

   8. associate-*r/ 97.6%

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

   9. add-sqr-sqrt 43.7%

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

   10. sqrt-unprod 41.9%

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

   11. sqr-neg 41.9%

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

   12. sqrt-unprod 16.8%

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

   13. add-sqr-sqrt 29.5%

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

   14. sub-neg 29.5%

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

   15. +-commutative 29.5%

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

   16. add-sqr-sqrt 12.6%

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

   17. sqrt-unprod 38.6%

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

   18. sqr-neg 38.6%

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

   19. sqrt-unprod 40.0%

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

   20. add-sqr-sqrt 21.5%

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

   21. sqrt-unprod 44.5%

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

   22. sqr-neg 44.5%

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

6. Applied egg-rr 97.6%

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

7. Taylor expanded in t1 around inf 60.8%

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

   1. mul-1-neg 60.8%

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

9. Simplified 60.8%

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

10. Final simplification 60.8%

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

Alternative 13: 14.1% accurate, 4.0× 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(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 62.4%

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

   1. associate-*l/ 68.0%

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

   2. *-commutative 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in t1 around inf 54.8%

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

   1. associate-*r/ 54.8%

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

   2. neg-mul-1 54.8%

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

7. Simplified 54.8%

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

   1. add-sqr-sqrt 35.9%

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

   2. sqrt-unprod 34.3%

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

   3. sqr-neg 34.3%

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

   4. sqrt-unprod 3.8%

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

   5. add-sqr-sqrt 10.7%

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

   6. div-inv 10.7%

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

9. Applied egg-rr 10.7%

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

    1. associate-*r/ 10.7%

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

    2. *-rgt-identity 10.7%

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

11. Simplified 10.7%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))