Rosa's DopplerBench

Percentage Accurate: 72.0% → 98.3%

Time: 9.3s

Alternatives: 12

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: 72.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. times-frac 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. distribute-neg-frac2 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. unsub-neg 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 87.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-u\right) - t1}\\ t_2 := t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -6.6 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-177}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 3.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;t1 \leq 2.65 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- u) t1))) (t_2 (* t1 (/ (- v) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -6.6e+66)
     t_1
     (if (<= t1 -1.05e-177)
       t_2
       (if (<= t1 3.3e-175)
         (/ (/ t1 u) (/ (- t1 u) v))
         (if (<= t1 2.65e+38) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-u - t1);
	double t_2 = t1 * (-v / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -6.6e+66) {
		tmp = t_1;
	} else if (t1 <= -1.05e-177) {
		tmp = t_2;
	} else if (t1 <= 3.3e-175) {
		tmp = (t1 / u) / ((t1 - u) / v);
	} else if (t1 <= 2.65e+38) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v / (-u - t1)
    t_2 = t1 * (-v / ((t1 + u) * (t1 + u)))
    if (t1 <= (-6.6d+66)) then
        tmp = t_1
    else if (t1 <= (-1.05d-177)) then
        tmp = t_2
    else if (t1 <= 3.3d-175) then
        tmp = (t1 / u) / ((t1 - u) / v)
    else if (t1 <= 2.65d+38) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-u - t1);
	double t_2 = t1 * (-v / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -6.6e+66) {
		tmp = t_1;
	} else if (t1 <= -1.05e-177) {
		tmp = t_2;
	} else if (t1 <= 3.3e-175) {
		tmp = (t1 / u) / ((t1 - u) / v);
	} else if (t1 <= 2.65e+38) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-u - t1)
	t_2 = t1 * (-v / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -6.6e+66:
		tmp = t_1
	elif t1 <= -1.05e-177:
		tmp = t_2
	elif t1 <= 3.3e-175:
		tmp = (t1 / u) / ((t1 - u) / v)
	elif t1 <= 2.65e+38:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-u) - t1))
	t_2 = Float64(t1 * Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -6.6e+66)
		tmp = t_1;
	elseif (t1 <= -1.05e-177)
		tmp = t_2;
	elseif (t1 <= 3.3e-175)
		tmp = Float64(Float64(t1 / u) / Float64(Float64(t1 - u) / v));
	elseif (t1 <= 2.65e+38)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-u - t1);
	t_2 = t1 * (-v / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -6.6e+66)
		tmp = t_1;
	elseif (t1 <= -1.05e-177)
		tmp = t_2;
	elseif (t1 <= 3.3e-175)
		tmp = (t1 / u) / ((t1 - u) / v);
	elseif (t1 <= 2.65e+38)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 * N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6.6e+66], t$95$1, If[LessEqual[t1, -1.05e-177], t$95$2, If[LessEqual[t1, 3.3e-175], N[(N[(t1 / u), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.65e+38], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -6.6000000000000003e66 or 2.65000000000000012e38 < t1

   1. Initial program 54.0%

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

      1. associate-/l* 45.8%

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

   3. Simplified 45.8%

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

      1. associate-*r/ 54.0%

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

      2. times-frac 100.0%

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

      3. frac-2neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. associate-*r/ 100.0%

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

      8. add-sqr-sqrt 50.7%

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

      9. sqrt-unprod 22.7%

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

      10. sqr-neg 22.7%

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

      11. sqrt-unprod 19.2%

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

      12. add-sqr-sqrt 32.4%

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

      13. sub-neg 32.4%

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

      14. +-commutative 32.4%

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

      15. add-sqr-sqrt 13.2%

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

      16. sqrt-unprod 48.1%

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

      17. sqr-neg 48.1%

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

      18. sqrt-unprod 45.8%

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

      19. add-sqr-sqrt 20.9%

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

      20. sqrt-unprod 43.8%

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

      21. sqr-neg 43.8%

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

      22. sqrt-unprod 24.9%

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

   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 92.2%

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

      1. mul-1-neg 92.2%

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

   9. Simplified 92.2%

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

   if -6.6000000000000003e66 < t1 < -1.05e-177 or 3.29999999999999999e-175 < t1 < 2.65000000000000012e38

   1. Initial program 91.2%

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

      1. associate-/l* 89.8%

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

   3. Simplified 89.8%

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

   if -1.05e-177 < t1 < 3.29999999999999999e-175

   1. Initial program 72.8%

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

      1. times-frac 95.3%

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

      2. distribute-frac-neg 95.3%

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

      3. distribute-neg-frac2 95.3%

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

      4. +-commutative 95.3%

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

      5. distribute-neg-in 95.3%

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

      6. unsub-neg 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in t1 around 0 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

      1. clear-num 86.8%

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

      2. un-div-inv 86.8%

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

      3. add-sqr-sqrt 35.9%

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

      4. sqrt-unprod 39.9%

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

      5. sqr-neg 39.9%

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

      6. sqrt-unprod 25.5%

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

      7. add-sqr-sqrt 42.7%

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

      8. frac-2neg 42.7%

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

      9. distribute-neg-in 42.7%

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

      10. add-sqr-sqrt 17.2%

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

      11. sqrt-unprod 39.1%

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

      12. sqr-neg 39.1%

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

      13. sqrt-unprod 23.7%

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

      14. add-sqr-sqrt 39.1%

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

      15. sub-neg 39.1%

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

      16. add-sqr-sqrt 27.1%

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

      17. sqrt-unprod 53.5%

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

      18. sqr-neg 53.5%

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

      19. sqrt-unprod 34.5%

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

      20. add-sqr-sqrt 90.4%

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

   9. Applied egg-rr 90.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-177}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 3.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;t1 \leq 2.65 \cdot 10^{+38}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{if}\;u \leq -7.6 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -1.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq -9.5 \cdot 10^{-6} \lor \neg \left(u \leq 2.36 \cdot 10^{+42}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (/ t1 u) u))))
   (if (<= u -7.6e+116)
     t_1
     (if (<= u -1.2e+71)
       (/ v (- (- u) t1))
       (if (or (<= u -9.5e-6) (not (<= u 2.36e+42))) t_1 (/ v (- t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / u);
	double tmp;
	if (u <= -7.6e+116) {
		tmp = t_1;
	} else if (u <= -1.2e+71) {
		tmp = v / (-u - t1);
	} else if ((u <= -9.5e-6) || !(u <= 2.36e+42)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = v * ((t1 / u) / u)
    if (u <= (-7.6d+116)) then
        tmp = t_1
    else if (u <= (-1.2d+71)) then
        tmp = v / (-u - t1)
    else if ((u <= (-9.5d-6)) .or. (.not. (u <= 2.36d+42))) then
        tmp = t_1
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / u);
	double tmp;
	if (u <= -7.6e+116) {
		tmp = t_1;
	} else if (u <= -1.2e+71) {
		tmp = v / (-u - t1);
	} else if ((u <= -9.5e-6) || !(u <= 2.36e+42)) {
		tmp = t_1;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * ((t1 / u) / u)
	tmp = 0
	if u <= -7.6e+116:
		tmp = t_1
	elif u <= -1.2e+71:
		tmp = v / (-u - t1)
	elif (u <= -9.5e-6) or not (u <= 2.36e+42):
		tmp = t_1
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(t1 / u) / u))
	tmp = 0.0
	if (u <= -7.6e+116)
		tmp = t_1;
	elseif (u <= -1.2e+71)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif ((u <= -9.5e-6) || !(u <= 2.36e+42))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * ((t1 / u) / u);
	tmp = 0.0;
	if (u <= -7.6e+116)
		tmp = t_1;
	elseif (u <= -1.2e+71)
		tmp = v / (-u - t1);
	elseif ((u <= -9.5e-6) || ~((u <= 2.36e+42)))
		tmp = t_1;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -7.6e+116], t$95$1, If[LessEqual[u, -1.2e+71], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -9.5e-6], N[Not[LessEqual[u, 2.36e+42]], $MachinePrecision]], t$95$1, N[(v / (-t1)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.5999999999999998e116 or -1.1999999999999999e71 < u < -9.5000000000000005e-6 or 2.36e42 < u

   1. Initial program 82.0%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 87.2%

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

      1. associate-*r/ 87.2%

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

      2. mul-1-neg 87.2%

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

   7. Simplified 87.2%

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

      1. associate-*l/ 88.8%

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

      2. frac-2neg 88.8%

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

      3. associate-*r/ 84.4%

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

      4. associate-*l/ 86.5%

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

      5. frac-2neg 86.5%

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

   9. Applied egg-rr 88.8%

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

   10. Taylor expanded in t1 around 0 84.4%

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

       1. mul-1-neg 84.4%

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

       2. *-commutative 84.4%

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

       3. associate-*r/ 86.4%

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

       4. distribute-rgt-neg-in 86.4%

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

   12. Simplified 86.4%

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

       1. associate-/l* 73.8%

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

       2. *-commutative 73.8%

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

       3. add-sqr-sqrt 48.2%

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

       4. sqrt-unprod 67.8%

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

       5. sqr-neg 67.8%

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

       6. sqrt-unprod 50.2%

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

       7. add-sqr-sqrt 66.0%

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

   14. Applied egg-rr 66.0%

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

   if -7.5999999999999998e116 < u < -1.1999999999999999e71

   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* 55.9%

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

   3. Simplified 55.9%

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

      1. associate-*r/ 54.5%

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

      2. times-frac 99.7%

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

      3. frac-2neg 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. sub-neg 99.7%

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

      7. associate-*r/ 100.0%

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

      8. add-sqr-sqrt 63.5%

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

      9. sqrt-unprod 29.2%

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

      10. sqr-neg 29.2%

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

      11. sqrt-unprod 9.7%

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

      12. add-sqr-sqrt 19.8%

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

      13. sub-neg 19.8%

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

      14. +-commutative 19.8%

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

      15. add-sqr-sqrt 10.1%

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

      16. sqrt-unprod 20.5%

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

      17. sqr-neg 20.5%

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

      18. sqrt-unprod 18.5%

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

      19. add-sqr-sqrt 18.5%

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

      20. sqrt-unprod 18.5%

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

      21. sqr-neg 18.5%

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

      22. sqrt-unprod 0.0%

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

   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 56.6%

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

      1. mul-1-neg 56.6%

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

   9. Simplified 56.6%

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

   if -9.5000000000000005e-6 < u < 2.36e42

   1. Initial program 66.6%

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

      1. associate-/l* 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in t1 around inf 73.3%

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

      1. associate-*r/ 73.3%

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

      2. neg-mul-1 73.3%

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

   7. Simplified 73.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+116}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq -1.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq -9.5 \cdot 10^{-6} \lor \neg \left(u \leq 2.36 \cdot 10^{+42}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2.65 \cdot 10^{+38}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -1.02e+170)
     (/ v (- t1))
     (if (<= t1 2.65e+38) (* 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 <= -1.02e+170) {
		tmp = v / -t1;
	} else if (t1 <= 2.65e+38) {
		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 <= (-1.02d+170)) then
        tmp = v / -t1
    else if (t1 <= 2.65d+38) 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 <= -1.02e+170) {
		tmp = v / -t1;
	} else if (t1 <= 2.65e+38) {
		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 <= -1.02e+170:
		tmp = v / -t1
	elif t1 <= 2.65e+38:
		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 <= -1.02e+170)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 2.65e+38)
		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 <= -1.02e+170)
		tmp = v / -t1;
	elseif (t1 <= 2.65e+38)
		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, -1.02e+170], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 2.65e+38], 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 < -1.02000000000000002e170

   1. Initial program 21.7%

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

      1. associate-/l* 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   if -1.02000000000000002e170 < t1 < 2.65000000000000012e38

   1. Initial program 83.2%

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

      1. associate-/l* 80.4%

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

   3. Simplified 80.4%

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

      1. associate-/r* 91.0%

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

      2. div-inv 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. associate-*r/ 91.0%

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

      2. *-rgt-identity 91.0%

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

   8. Simplified 91.0%

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

   if 2.65000000000000012e38 < t1

   1. Initial program 71.2%

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

      1. associate-/l* 56.4%

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

   3. Simplified 56.4%

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

      1. associate-*r/ 71.2%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 14.9%

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

      10. sqr-neg 14.9%

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

      11. sqrt-unprod 39.2%

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

      12. add-sqr-sqrt 39.2%

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

      13. sub-neg 39.2%

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

      14. +-commutative 39.2%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 70.2%

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

      17. sqr-neg 70.2%

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

      18. sqrt-unprod 93.5%

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

      19. add-sqr-sqrt 42.7%

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

      20. sqrt-unprod 89.5%

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

      21. sqr-neg 89.5%

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

      22. sqrt-unprod 50.8%

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

   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 90.1%

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

      1. mul-1-neg 90.1%

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

   9. Simplified 90.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2.65 \cdot 10^{+38}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.5e+40)
   (/ v (- (- u) t1))
   (if (<= t1 4.6e-57)
     (/ (/ (- t1) u) (/ u v))
     (* (/ v (+ t1 u)) (+ (/ u t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.5e+40) {
		tmp = v / (-u - t1);
	} else if (t1 <= 4.6e-57) {
		tmp = (-t1 / u) / (u / v);
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	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 <= (-2.5d+40)) then
        tmp = v / (-u - t1)
    else if (t1 <= 4.6d-57) then
        tmp = (-t1 / u) / (u / v)
    else
        tmp = (v / (t1 + u)) * ((u / t1) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.5e+40) {
		tmp = v / (-u - t1);
	} else if (t1 <= 4.6e-57) {
		tmp = (-t1 / u) / (u / v);
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.5e+40:
		tmp = v / (-u - t1)
	elif t1 <= 4.6e-57:
		tmp = (-t1 / u) / (u / v)
	else:
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.5e+40)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 4.6e-57)
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u / t1) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.5e+40)
		tmp = v / (-u - t1);
	elseif (t1 <= 4.6e-57)
		tmp = (-t1 / u) / (u / v);
	else
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.5e+40], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.6e-57], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.50000000000000002e40

   1. Initial program 42.2%

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

      1. associate-/l* 40.6%

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

   3. Simplified 40.6%

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

      1. associate-*r/ 42.2%

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

      2. times-frac 100.0%

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

      3. frac-2neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. associate-*r/ 100.0%

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

      8. add-sqr-sqrt 99.4%

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

      9. sqrt-unprod 35.4%

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

      10. sqr-neg 35.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 27.8%

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

      13. sub-neg 27.8%

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

      14. +-commutative 27.8%

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

      15. add-sqr-sqrt 27.8%

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

      16. sqrt-unprod 28.6%

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

      17. sqr-neg 28.6%

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

      18. sqrt-unprod 0.0%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 0.0%

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

      21. sqr-neg 0.0%

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

      22. sqrt-unprod 0.0%

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

   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 94.7%

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

      1. mul-1-neg 94.7%

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

   9. Simplified 94.7%

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

   if -2.50000000000000002e40 < t1 < 4.6e-57

   1. Initial program 82.3%

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

      1. times-frac 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-neg-frac2 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

   7. Simplified 78.8%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.9%

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

      3. add-sqr-sqrt 40.8%

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

      4. sqrt-unprod 50.7%

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

      5. sqr-neg 50.7%

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

      6. sqrt-unprod 22.4%

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

      7. add-sqr-sqrt 43.3%

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

      8. frac-2neg 43.3%

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

      9. distribute-neg-in 43.3%

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

      10. add-sqr-sqrt 20.9%

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

      11. sqrt-unprod 40.4%

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

      12. sqr-neg 40.4%

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

      13. sqrt-unprod 21.7%

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

      14. add-sqr-sqrt 41.7%

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

      15. sub-neg 41.7%

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

      16. add-sqr-sqrt 26.9%

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

      17. sqrt-unprod 53.2%

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

      18. sqr-neg 53.2%

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

      19. sqrt-unprod 35.9%

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

      20. add-sqr-sqrt 80.5%

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

   9. Applied egg-rr 80.5%

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

   10. Taylor expanded in t1 around 0 82.1%

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

       1. neg-mul-1 82.1%

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

       2. distribute-neg-frac2 82.1%

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

   12. Simplified 82.1%

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

   if 4.6e-57 < t1

   1. Initial program 77.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 inf 85.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+42} \lor \neg \left(t1 \leq 3.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.2e+42) (not (<= t1 3.4e-57)))
   (/ v (- (- u) t1))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e+42) || !(t1 <= 3.4e-57)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.2d+42)) .or. (.not. (t1 <= 3.4d-57))) then
        tmp = v / (-u - t1)
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e+42) || !(t1 <= 3.4e-57)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.2e+42) or not (t1 <= 3.4e-57):
		tmp = v / (-u - t1)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.2e+42) || !(t1 <= 3.4e-57))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.2e+42) || ~((t1 <= 3.4e-57)))
		tmp = v / (-u - t1);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.2e+42], N[Not[LessEqual[t1, 3.4e-57]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.2000000000000001e42 or 3.40000000000000016e-57 < t1

   1. Initial program 60.8%

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

      1. associate-/l* 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*r/ 60.8%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 100.0%

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

      8. add-sqr-sqrt 46.6%

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

      9. sqrt-unprod 26.5%

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

      10. sqr-neg 26.5%

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

      11. sqrt-unprod 20.0%

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

      12. add-sqr-sqrt 33.0%

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

      13. sub-neg 33.0%

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

      14. +-commutative 33.0%

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

      15. add-sqr-sqrt 13.0%

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

      16. sqrt-unprod 51.9%

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

      17. sqr-neg 51.9%

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

      18. sqrt-unprod 48.1%

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

      19. add-sqr-sqrt 22.2%

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

      20. sqrt-unprod 47.0%

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

      21. sqr-neg 47.0%

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

      22. sqrt-unprod 26.4%

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

   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 89.5%

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

      1. mul-1-neg 89.5%

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

   9. Simplified 89.5%

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

   if -2.2000000000000001e42 < t1 < 3.40000000000000016e-57

   1. Initial program 82.3%

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

      1. times-frac 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-neg-frac2 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

   7. Simplified 78.8%

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

      1. associate-*l/ 77.4%

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

      2. frac-2neg 77.4%

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

      3. associate-*r/ 71.1%

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

      4. associate-*l/ 76.9%

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

      5. frac-2neg 76.9%

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

   9. Applied egg-rr 78.9%

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

   10. Taylor expanded in t1 around 0 74.3%

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

       1. mul-1-neg 74.3%

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

       2. *-commutative 74.3%

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

       3. associate-*r/ 80.1%

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

       4. distribute-rgt-neg-in 80.1%

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

   12. Simplified 80.1%

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

       1. frac-2neg 80.1%

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

       2. distribute-frac-neg 80.1%

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

       3. add-sqr-sqrt 44.8%

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

       4. sqrt-unprod 50.3%

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

       5. sqr-neg 50.3%

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

       6. sqrt-unprod 33.8%

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

       7. add-sqr-sqrt 39.9%

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

       8. remove-double-neg 39.9%

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

       9. distribute-rgt-neg-out 39.9%

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

       10. frac-2neg 39.9%

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

       11. *-commutative 39.9%

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

       12. associate-/l* 39.9%

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

       13. add-sqr-sqrt 30.7%

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

       14. sqrt-unprod 56.1%

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

       15. sqr-neg 56.1%

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

       16. sqrt-unprod 60.4%

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

       17. add-sqr-sqrt 82.0%

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

   14. Applied egg-rr 82.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+42} \lor \neg \left(t1 \leq 3.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+40} \lor \neg \left(t1 \leq 4.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.5e+40) (not (<= t1 4.4e-57)))
   (/ v (- (- u) t1))
   (/ (/ (- t1) u) (/ u v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.5e+40) || !(t1 <= 4.4e-57)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	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 <= (-2.5d+40)) .or. (.not. (t1 <= 4.4d-57))) then
        tmp = v / (-u - t1)
    else
        tmp = (-t1 / u) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.5e+40) || !(t1 <= 4.4e-57)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.5e+40) or not (t1 <= 4.4e-57):
		tmp = v / (-u - t1)
	else:
		tmp = (-t1 / u) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.5e+40) || !(t1 <= 4.4e-57))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.5e+40) || ~((t1 <= 4.4e-57)))
		tmp = v / (-u - t1);
	else
		tmp = (-t1 / u) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.5e+40], N[Not[LessEqual[t1, 4.4e-57]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.50000000000000002e40 or 4.39999999999999997e-57 < t1

   1. Initial program 60.8%

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

      1. associate-/l* 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*r/ 60.8%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 100.0%

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

      8. add-sqr-sqrt 46.6%

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

      9. sqrt-unprod 26.5%

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

      10. sqr-neg 26.5%

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

      11. sqrt-unprod 20.0%

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

      12. add-sqr-sqrt 33.0%

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

      13. sub-neg 33.0%

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

      14. +-commutative 33.0%

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

      15. add-sqr-sqrt 13.0%

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

      16. sqrt-unprod 51.9%

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

      17. sqr-neg 51.9%

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

      18. sqrt-unprod 48.1%

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

      19. add-sqr-sqrt 22.2%

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

      20. sqrt-unprod 47.0%

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

      21. sqr-neg 47.0%

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

      22. sqrt-unprod 26.4%

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

   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 89.5%

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

      1. mul-1-neg 89.5%

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

   9. Simplified 89.5%

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

   if -2.50000000000000002e40 < t1 < 4.39999999999999997e-57

   1. Initial program 82.3%

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

      1. times-frac 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-neg-frac2 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

   7. Simplified 78.8%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.9%

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

      3. add-sqr-sqrt 40.8%

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

      4. sqrt-unprod 50.7%

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

      5. sqr-neg 50.7%

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

      6. sqrt-unprod 22.4%

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

      7. add-sqr-sqrt 43.3%

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

      8. frac-2neg 43.3%

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

      9. distribute-neg-in 43.3%

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

      10. add-sqr-sqrt 20.9%

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

      11. sqrt-unprod 40.4%

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

      12. sqr-neg 40.4%

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

      13. sqrt-unprod 21.7%

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

      14. add-sqr-sqrt 41.7%

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

      15. sub-neg 41.7%

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

      16. add-sqr-sqrt 26.9%

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

      17. sqrt-unprod 53.2%

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

      18. sqr-neg 53.2%

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

      19. sqrt-unprod 35.9%

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

      20. add-sqr-sqrt 80.5%

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

   9. Applied egg-rr 80.5%

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

   10. Taylor expanded in t1 around 0 82.1%

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

       1. neg-mul-1 82.1%

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

       2. distribute-neg-frac2 82.1%

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

   12. Simplified 82.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+40} \lor \neg \left(t1 \leq 4.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+117} \lor \neg \left(u \leq 4.4 \cdot 10^{+215}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.9e+117) (not (<= u 4.4e+215))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.9e+117) || !(u <= 4.4e+215)) {
		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 <= (-4.9d+117)) .or. (.not. (u <= 4.4d+215))) 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 <= -4.9e+117) || !(u <= 4.4e+215)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.9e+117) or not (u <= 4.4e+215):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.9e+117) || !(u <= 4.4e+215))
		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 <= -4.9e+117) || ~((u <= 4.4e+215)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.9e+117], N[Not[LessEqual[u, 4.4e+215]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.9000000000000001e117 or 4.4000000000000003e215 < u

   1. Initial program 84.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.2%

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

      1. associate-*r/ 98.2%

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

      2. mul-1-neg 98.2%

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

   7. Simplified 98.2%

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

      1. associate-*l/ 98.2%

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

      2. frac-2neg 98.2%

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

      3. associate-*r/ 91.0%

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

      4. associate-*l/ 98.2%

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

      5. frac-2neg 98.2%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in t1 around inf 44.8%

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

   if -4.9000000000000001e117 < u < 4.4000000000000003e215

   1. Initial program 69.8%

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

      1. associate-/l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in t1 around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+117} \lor \neg \left(u \leq 4.4 \cdot 10^{+215}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9.6e+117) (/ v (- u)) (if (<= u 4.1e+215) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.6e+117) {
		tmp = v / -u;
	} else if (u <= 4.1e+215) {
		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 (u <= (-9.6d+117)) then
        tmp = v / -u
    else if (u <= 4.1d+215) 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 (u <= -9.6e+117) {
		tmp = v / -u;
	} else if (u <= 4.1e+215) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9.6e+117:
		tmp = v / -u
	elif u <= 4.1e+215:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9.6e+117)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 4.1e+215)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9.6e+117)
		tmp = v / -u;
	elseif (u <= 4.1e+215)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9.6e+117], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 4.1e+215], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -9.5999999999999996e117

   1. Initial program 82.7%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 97.5%

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

      1. associate-*r/ 97.5%

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

      2. mul-1-neg 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in t1 around inf 40.7%

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

      1. associate-*r/ 40.7%

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

      2. mul-1-neg 40.7%

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

   10. Simplified 40.7%

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

   if -9.5999999999999996e117 < u < 4.1000000000000004e215

   1. Initial program 69.8%

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

      1. associate-/l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in t1 around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   if 4.1000000000000004e215 < u

   1. Initial program 87.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9%

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

      1. associate-*r/ 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*l/ 100.0%

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

      2. frac-2neg 100.0%

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

      3. associate-*r/ 93.5%

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

      4. associate-*l/ 99.9%

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

      5. frac-2neg 99.9%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in t1 around inf 55.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+131} \lor \neg \left(t1 \leq 1.35 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.4e+131) (not (<= t1 1.35e+123))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.4e+131) || !(t1 <= 1.35e+123)) {
		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 <= (-1.4d+131)) .or. (.not. (t1 <= 1.35d+123))) 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 <= -1.4e+131) || !(t1 <= 1.35e+123)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.4e+131) or not (t1 <= 1.35e+123):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.4e+131) || !(t1 <= 1.35e+123))
		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 <= -1.4e+131) || ~((t1 <= 1.35e+123)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.4e+131], N[Not[LessEqual[t1, 1.35e+123]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4e131 or 1.35000000000000007e123 < t1

   1. Initial program 41.5%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 94.6%

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

   6. Taylor expanded in u around inf 29.4%

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

   if -1.4e131 < t1 < 1.35000000000000007e123

   1. Initial program 85.1%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. mul-1-neg 71.4%

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

   7. Simplified 71.4%

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

      1. associate-*l/ 70.3%

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

      2. frac-2neg 70.3%

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

      3. associate-*r/ 64.4%

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

      4. associate-*l/ 69.4%

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

      5. frac-2neg 69.4%

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

   9. Applied egg-rr 71.6%

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

   10. Taylor expanded in t1 around inf 20.2%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+131} \lor \neg \left(t1 \leq 1.35 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% 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 72.8%

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

   1. associate-/l* 68.4%

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

3. Simplified 68.4%

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

   1. associate-*r/ 72.8%

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

   2. times-frac 98.8%

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

   3. frac-2neg 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. sub-neg 98.8%

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

   7. associate-*r/ 97.7%

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

   8. add-sqr-sqrt 50.2%

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

   9. sqrt-unprod 45.2%

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

   10. sqr-neg 45.2%

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

   11. sqrt-unprod 20.9%

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

   12. add-sqr-sqrt 36.9%

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

   13. sub-neg 36.9%

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

   14. +-commutative 36.9%

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

   15. add-sqr-sqrt 16.0%

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

   16. sqrt-unprod 49.0%

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

   17. sqr-neg 49.0%

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

   18. sqrt-unprod 36.4%

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

   19. add-sqr-sqrt 17.3%

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

   20. sqrt-unprod 41.0%

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

   21. sqr-neg 41.0%

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

   22. sqrt-unprod 25.7%

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

6. Applied egg-rr 97.7%

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

7. Taylor expanded in t1 around inf 59.3%

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

   1. mul-1-neg 59.3%

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

9. Simplified 59.3%

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

10. Final simplification 59.3%

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

Alternative 12: 14.0% 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 72.8%

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

   1. times-frac 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. distribute-neg-frac2 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. unsub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in t1 around inf 52.0%

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

6. Taylor expanded in u around inf 11.0%

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

7. Final simplification 11.0%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))