Rosa's DopplerBench

Percentage Accurate: 72.7% → 97.8%

Time: 11.7s

Alternatives: 19

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: 72.7% 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: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-v}{t1 + u}}{\frac{u}{t1} + 1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (/ (- v) (+ t1 u)) (+ (/ u t1) 1.0)))

C

double code(double u, double v, double t1) {
	return (-v / (t1 + u)) / ((u / t1) + 1.0);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-v / (t1 + u)) / ((u / t1) + 1.0d0)
end function

Java

public static double code(double u, double v, double t1) {
	return (-v / (t1 + u)) / ((u / t1) + 1.0);
}

Python

def code(u, v, t1):
	return (-v / (t1 + u)) / ((u / t1) + 1.0)

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-v) / Float64(t1 + u)) / Float64(Float64(u / t1) + 1.0))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-v / (t1 + u)) / ((u / t1) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 73.2%

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

   1. times-frac 97.6%

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

   2. neg-mul-1 97.6%

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

   3. associate-/l* 97.6%

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

   4. associate-*l/ 97.7%

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

   5. neg-mul-1 97.7%

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

   6. distribute-frac-neg 97.7%

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

   7. +-commutative 97.7%

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

   8. remove-double-neg 97.7%

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

   9. unsub-neg 97.7%

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

   10. div-sub 97.7%

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

   11. sub-neg 97.7%

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

   12. distribute-frac-neg 97.7%

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

   13. remove-double-neg 97.7%

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

   14. *-inverses 97.7%

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

3. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 2: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{if}\;u \leq -8.6 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.45 \cdot 10^{-48}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 68000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))) (t_2 (/ t1 (* u (/ (- t1 u) v)))))
   (if (<= u -8.6e-34)
     t_2
     (if (<= u 4.5e-73)
       t_1
       (if (<= u 2.45e-48)
         (* v (/ (/ t1 u) (- t1 u)))
         (if (<= u 68000000000000.0)
           t_1
           (if (<= u 5.6e+152) t_2 (/ (* t1 (/ v u)) (- u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = t1 / (u * ((t1 - u) / v));
	double tmp;
	if (u <= -8.6e-34) {
		tmp = t_2;
	} else if (u <= 4.5e-73) {
		tmp = t_1;
	} else if (u <= 2.45e-48) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 68000000000000.0) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t_2;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = t1 / (u * ((t1 - u) / v))
    if (u <= (-8.6d-34)) then
        tmp = t_2
    else if (u <= 4.5d-73) then
        tmp = t_1
    else if (u <= 2.45d-48) then
        tmp = v * ((t1 / u) / (t1 - u))
    else if (u <= 68000000000000.0d0) then
        tmp = t_1
    else if (u <= 5.6d+152) then
        tmp = t_2
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = t1 / (u * ((t1 - u) / v));
	double tmp;
	if (u <= -8.6e-34) {
		tmp = t_2;
	} else if (u <= 4.5e-73) {
		tmp = t_1;
	} else if (u <= 2.45e-48) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 68000000000000.0) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t_2;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = t1 / (u * ((t1 - u) / v))
	tmp = 0
	if u <= -8.6e-34:
		tmp = t_2
	elif u <= 4.5e-73:
		tmp = t_1
	elif u <= 2.45e-48:
		tmp = v * ((t1 / u) / (t1 - u))
	elif u <= 68000000000000.0:
		tmp = t_1
	elif u <= 5.6e+152:
		tmp = t_2
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)))
	tmp = 0.0
	if (u <= -8.6e-34)
		tmp = t_2;
	elseif (u <= 4.5e-73)
		tmp = t_1;
	elseif (u <= 2.45e-48)
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)));
	elseif (u <= 68000000000000.0)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = t_2;
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = t1 / (u * ((t1 - u) / v));
	tmp = 0.0;
	if (u <= -8.6e-34)
		tmp = t_2;
	elseif (u <= 4.5e-73)
		tmp = t_1;
	elseif (u <= 2.45e-48)
		tmp = v * ((t1 / u) / (t1 - u));
	elseif (u <= 68000000000000.0)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = t_2;
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -8.6e-34], t$95$2, If[LessEqual[u, 4.5e-73], t$95$1, If[LessEqual[u, 2.45e-48], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 68000000000000.0], t$95$1, If[LessEqual[u, 5.6e+152], t$95$2, N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -8.5999999999999999e-34 or 6.8e13 < u < 5.6000000000000004e152

   1. Initial program 83.0%

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

      1. times-frac 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in t1 around 0 87.9%

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

      1. mul-1-neg 87.9%

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

      2. distribute-neg-frac 87.9%

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

   6. Simplified 87.9%

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

      1. *-commutative 87.9%

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

      2. clear-num 88.5%

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

      3. frac-times 87.6%

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

      4. *-un-lft-identity 87.6%

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

      5. add-sqr-sqrt 52.2%

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

      6. sqrt-unprod 59.8%

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

      7. sqr-neg 59.8%

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

      8. sqrt-unprod 22.2%

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

      9. add-sqr-sqrt 55.6%

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

      10. frac-2neg 55.6%

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

      11. add-sqr-sqrt 26.9%

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

      12. sqrt-unprod 65.5%

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

      13. sqr-neg 65.5%

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

      14. sqrt-unprod 49.3%

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

      15. add-sqr-sqrt 87.6%

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

      16. distribute-neg-in 87.6%

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

      17. add-sqr-sqrt 52.3%

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

      18. sqrt-unprod 86.5%

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

      19. sqr-neg 86.5%

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

      20. sqrt-unprod 35.3%

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

      21. add-sqr-sqrt 87.4%

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

      22. sub-neg 87.4%

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

   8. Applied egg-rr 87.4%

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

   if -8.5999999999999999e-34 < u < 4.5e-73 or 2.4500000000000001e-48 < u < 6.8e13

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 4.5e-73 < u < 2.4500000000000001e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

      1. clear-num 86.5%

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

      2. frac-times 86.3%

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

      3. *-commutative 86.3%

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

      4. *-un-lft-identity 86.3%

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

      5. add-sqr-sqrt 54.7%

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

      6. sqrt-unprod 30.2%

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

      7. sqr-neg 30.2%

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

      8. sqrt-unprod 1.1%

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

      9. add-sqr-sqrt 13.3%

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

      10. frac-2neg 13.3%

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

      11. add-sqr-sqrt 1.2%

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

      12. sqrt-unprod 43.5%

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

      13. sqr-neg 43.5%

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

      14. sqrt-unprod 42.0%

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

      15. add-sqr-sqrt 86.3%

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

      16. distribute-neg-in 86.3%

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

      17. add-sqr-sqrt 55.2%

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

      18. sqrt-unprod 86.0%

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

      19. sqr-neg 86.0%

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

      20. sqrt-unprod 30.8%

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

      21. add-sqr-sqrt 86.0%

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

      22. sub-neg 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. associate-/r* 85.9%

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

      2. associate-/r/ 96.5%

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

   10. Simplified 96.5%

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

   if 5.6000000000000004e152 < u

   1. Initial program 71.6%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around 0 96.4%

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

      1. mul-1-neg 96.4%

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

      2. distribute-neg-frac 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in t1 around 0 96.4%

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

      1. frac-2neg 96.4%

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

      2. remove-double-neg 96.4%

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

      3. associate-*l/ 99.1%

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

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u}}{-u}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 2.45 \cdot 10^{-48}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 68000000000000:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 3: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{if}\;u \leq -1 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))) (t_2 (/ t1 (* u (/ (- t1 u) v)))))
   (if (<= u -1e-33)
     t_2
     (if (<= u 8.2e-72)
       t_1
       (if (<= u 1.8e-48)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= u 1.2e+29)
           t_1
           (if (<= u 5.6e+152) t_2 (/ (* t1 (/ v u)) (- u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = t1 / (u * ((t1 - u) / v));
	double tmp;
	if (u <= -1e-33) {
		tmp = t_2;
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 1.8e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 1.2e+29) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t_2;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = t1 / (u * ((t1 - u) / v))
    if (u <= (-1d-33)) then
        tmp = t_2
    else if (u <= 8.2d-72) then
        tmp = t_1
    else if (u <= 1.8d-48) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 1.2d+29) then
        tmp = t_1
    else if (u <= 5.6d+152) then
        tmp = t_2
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = t1 / (u * ((t1 - u) / v));
	double tmp;
	if (u <= -1e-33) {
		tmp = t_2;
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 1.8e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 1.2e+29) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t_2;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = t1 / (u * ((t1 - u) / v))
	tmp = 0
	if u <= -1e-33:
		tmp = t_2
	elif u <= 8.2e-72:
		tmp = t_1
	elif u <= 1.8e-48:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 1.2e+29:
		tmp = t_1
	elif u <= 5.6e+152:
		tmp = t_2
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)))
	tmp = 0.0
	if (u <= -1e-33)
		tmp = t_2;
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 1.8e-48)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 1.2e+29)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = t_2;
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = t1 / (u * ((t1 - u) / v));
	tmp = 0.0;
	if (u <= -1e-33)
		tmp = t_2;
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 1.8e-48)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 1.2e+29)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = t_2;
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1e-33], t$95$2, If[LessEqual[u, 8.2e-72], t$95$1, If[LessEqual[u, 1.8e-48], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.2e+29], t$95$1, If[LessEqual[u, 5.6e+152], t$95$2, N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.0000000000000001e-33 or 1.2e29 < u < 5.6000000000000004e152

   1. Initial program 83.0%

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

      1. times-frac 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in t1 around 0 87.9%

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

      1. mul-1-neg 87.9%

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

      2. distribute-neg-frac 87.9%

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

   6. Simplified 87.9%

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

      1. *-commutative 87.9%

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

      2. clear-num 88.5%

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

      3. frac-times 87.6%

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

      4. *-un-lft-identity 87.6%

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

      5. add-sqr-sqrt 52.2%

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

      6. sqrt-unprod 59.8%

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

      7. sqr-neg 59.8%

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

      8. sqrt-unprod 22.2%

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

      9. add-sqr-sqrt 55.6%

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

      10. frac-2neg 55.6%

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

      11. add-sqr-sqrt 26.9%

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

      12. sqrt-unprod 65.5%

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

      13. sqr-neg 65.5%

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

      14. sqrt-unprod 49.3%

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

      15. add-sqr-sqrt 87.6%

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

      16. distribute-neg-in 87.6%

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

      17. add-sqr-sqrt 52.3%

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

      18. sqrt-unprod 86.5%

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

      19. sqr-neg 86.5%

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

      20. sqrt-unprod 35.3%

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

      21. add-sqr-sqrt 87.4%

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

      22. sub-neg 87.4%

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

   8. Applied egg-rr 87.4%

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

   if -1.0000000000000001e-33 < u < 8.20000000000000007e-72 or 1.8000000000000001e-48 < u < 1.2e29

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 8.20000000000000007e-72 < u < 1.8000000000000001e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

      1. expm1-log1p-u 55.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

   8. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 55.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)\right)} \]

      2. expm1-log1p 96.6%

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

   10. Simplified 96.6%

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

   if 5.6000000000000004e152 < u

   1. Initial program 71.6%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around 0 96.4%

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

      1. mul-1-neg 96.4%

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

      2. distribute-neg-frac 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in t1 around 0 96.4%

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

      1. frac-2neg 96.4%

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

      2. remove-double-neg 96.4%

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

      3. associate-*l/ 99.1%

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

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u}}{-u}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 4: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{if}\;u \leq -2.2 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 75000000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0))))
        (t_2 (/ (* t1 (/ v (+ t1 u))) (- t1 u))))
   (if (<= u -2.2e-34)
     t_2
     (if (<= u 8.2e-72)
       t_1
       (if (<= u 2.8e-48)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= u 75000000000000.0) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 * (v / (t1 + u))) / (t1 - u);
	double tmp;
	if (u <= -2.2e-34) {
		tmp = t_2;
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 2.8e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 75000000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = (t1 * (v / (t1 + u))) / (t1 - u)
    if (u <= (-2.2d-34)) then
        tmp = t_2
    else if (u <= 8.2d-72) then
        tmp = t_1
    else if (u <= 2.8d-48) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 75000000000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 * (v / (t1 + u))) / (t1 - u);
	double tmp;
	if (u <= -2.2e-34) {
		tmp = t_2;
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 2.8e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 75000000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = (t1 * (v / (t1 + u))) / (t1 - u)
	tmp = 0
	if u <= -2.2e-34:
		tmp = t_2
	elif u <= 8.2e-72:
		tmp = t_1
	elif u <= 2.8e-48:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 75000000000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(t1 - u))
	tmp = 0.0
	if (u <= -2.2e-34)
		tmp = t_2;
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 2.8e-48)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 75000000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = (t1 * (v / (t1 + u))) / (t1 - u);
	tmp = 0.0;
	if (u <= -2.2e-34)
		tmp = t_2;
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 2.8e-48)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 75000000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.2e-34], t$95$2, If[LessEqual[u, 8.2e-72], t$95$1, If[LessEqual[u, 2.8e-48], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 75000000000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1999999999999999e-34 or 7.5e13 < u

   1. Initial program 80.0%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

      1. frac-2neg 98.3%

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

      2. remove-double-neg 98.3%

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

      3. associate-*l/ 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. add-sqr-sqrt 57.7%

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

      6. sqrt-unprod 90.2%

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

      7. sqr-neg 90.2%

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

      8. sqrt-unprod 40.2%

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

      9. add-sqr-sqrt 91.8%

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

      10. sub-neg 91.8%

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

   5. Applied egg-rr 91.8%

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

   if -2.1999999999999999e-34 < u < 8.20000000000000007e-72 or 2.80000000000000005e-48 < u < 7.5e13

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 8.20000000000000007e-72 < u < 2.80000000000000005e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

      1. expm1-log1p-u 55.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

   8. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 55.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)\right)} \]

      2. expm1-log1p 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 75000000000000:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \end{array} \]

Alternative 5: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{if}\;u \leq -1.9 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 7 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.45 \cdot 10^{-48}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))) (t_2 (/ (* t1 (/ v u)) (- u))))
   (if (<= u -1.9e-32)
     t_2
     (if (<= u 7e-72)
       t_1
       (if (<= u 2.45e-48)
         (* (/ t1 u) (/ (- v) u))
         (if (<= u 6.5e+19)
           t_1
           (if (<= u 5.6e+152) (/ t1 (* (- u) (/ u v))) t_2)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -1.9e-32) {
		tmp = t_2;
	} else if (u <= 7e-72) {
		tmp = t_1;
	} else if (u <= 2.45e-48) {
		tmp = (t1 / u) * (-v / u);
	} else if (u <= 6.5e+19) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t1 / (-u * (u / v));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = (t1 * (v / u)) / -u
    if (u <= (-1.9d-32)) then
        tmp = t_2
    else if (u <= 7d-72) then
        tmp = t_1
    else if (u <= 2.45d-48) then
        tmp = (t1 / u) * (-v / u)
    else if (u <= 6.5d+19) then
        tmp = t_1
    else if (u <= 5.6d+152) then
        tmp = t1 / (-u * (u / v))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -1.9e-32) {
		tmp = t_2;
	} else if (u <= 7e-72) {
		tmp = t_1;
	} else if (u <= 2.45e-48) {
		tmp = (t1 / u) * (-v / u);
	} else if (u <= 6.5e+19) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t1 / (-u * (u / v));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = (t1 * (v / u)) / -u
	tmp = 0
	if u <= -1.9e-32:
		tmp = t_2
	elif u <= 7e-72:
		tmp = t_1
	elif u <= 2.45e-48:
		tmp = (t1 / u) * (-v / u)
	elif u <= 6.5e+19:
		tmp = t_1
	elif u <= 5.6e+152:
		tmp = t1 / (-u * (u / v))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(Float64(t1 * Float64(v / u)) / Float64(-u))
	tmp = 0.0
	if (u <= -1.9e-32)
		tmp = t_2;
	elseif (u <= 7e-72)
		tmp = t_1;
	elseif (u <= 2.45e-48)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif (u <= 6.5e+19)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = Float64(t1 / Float64(Float64(-u) * Float64(u / v)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = (t1 * (v / u)) / -u;
	tmp = 0.0;
	if (u <= -1.9e-32)
		tmp = t_2;
	elseif (u <= 7e-72)
		tmp = t_1;
	elseif (u <= 2.45e-48)
		tmp = (t1 / u) * (-v / u);
	elseif (u <= 6.5e+19)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = t1 / (-u * (u / v));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -1.9e-32], t$95$2, If[LessEqual[u, 7e-72], t$95$1, If[LessEqual[u, 2.45e-48], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.5e+19], t$95$1, If[LessEqual[u, 5.6e+152], N[(t1 / N[((-u) * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.90000000000000004e-32 or 5.6000000000000004e152 < u

   1. Initial program 75.4%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around 0 88.7%

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

      1. mul-1-neg 88.7%

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

      2. distribute-neg-frac 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in t1 around 0 86.7%

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

      1. frac-2neg 86.7%

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

      2. remove-double-neg 86.7%

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

      3. associate-*l/ 87.7%

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

   9. Applied egg-rr 87.7%

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

   if -1.90000000000000004e-32 < u < 7.00000000000000001e-72 or 2.4500000000000001e-48 < u < 6.5e19

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 7.00000000000000001e-72 < u < 2.4500000000000001e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in t1 around 0 86.3%

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

   if 6.5e19 < u < 5.6000000000000004e152

   1. Initial program 99.9%

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

      1. times-frac 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around 0 96.2%

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

      1. mul-1-neg 96.2%

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

      2. distribute-neg-frac 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in t1 around 0 76.2%

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

      1. *-commutative 76.2%

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

      2. clear-num 76.2%

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

      3. frac-2neg 76.2%

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

      4. frac-times 92.0%

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

      5. *-un-lft-identity 92.0%

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

      6. remove-double-neg 92.0%

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

   9. Applied egg-rr 92.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 7 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 2.45 \cdot 10^{-48}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 6: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{if}\;u \leq -3.7 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))) (t_2 (/ (* t1 (/ v u)) (- u))))
   (if (<= u -3.7e-32)
     t_2
     (if (<= u 5.2e-73)
       t_1
       (if (<= u 1.8e-48)
         (* v (/ (/ t1 u) (- t1 u)))
         (if (<= u 2.6e+18)
           t_1
           (if (<= u 5.6e+152) (/ t1 (* (- u) (/ u v))) t_2)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -3.7e-32) {
		tmp = t_2;
	} else if (u <= 5.2e-73) {
		tmp = t_1;
	} else if (u <= 1.8e-48) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 2.6e+18) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t1 / (-u * (u / v));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = (t1 * (v / u)) / -u
    if (u <= (-3.7d-32)) then
        tmp = t_2
    else if (u <= 5.2d-73) then
        tmp = t_1
    else if (u <= 1.8d-48) then
        tmp = v * ((t1 / u) / (t1 - u))
    else if (u <= 2.6d+18) then
        tmp = t_1
    else if (u <= 5.6d+152) then
        tmp = t1 / (-u * (u / v))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -3.7e-32) {
		tmp = t_2;
	} else if (u <= 5.2e-73) {
		tmp = t_1;
	} else if (u <= 1.8e-48) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 2.6e+18) {
		tmp = t_1;
	} else if (u <= 5.6e+152) {
		tmp = t1 / (-u * (u / v));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = (t1 * (v / u)) / -u
	tmp = 0
	if u <= -3.7e-32:
		tmp = t_2
	elif u <= 5.2e-73:
		tmp = t_1
	elif u <= 1.8e-48:
		tmp = v * ((t1 / u) / (t1 - u))
	elif u <= 2.6e+18:
		tmp = t_1
	elif u <= 5.6e+152:
		tmp = t1 / (-u * (u / v))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(Float64(t1 * Float64(v / u)) / Float64(-u))
	tmp = 0.0
	if (u <= -3.7e-32)
		tmp = t_2;
	elseif (u <= 5.2e-73)
		tmp = t_1;
	elseif (u <= 1.8e-48)
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)));
	elseif (u <= 2.6e+18)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = Float64(t1 / Float64(Float64(-u) * Float64(u / v)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = (t1 * (v / u)) / -u;
	tmp = 0.0;
	if (u <= -3.7e-32)
		tmp = t_2;
	elseif (u <= 5.2e-73)
		tmp = t_1;
	elseif (u <= 1.8e-48)
		tmp = v * ((t1 / u) / (t1 - u));
	elseif (u <= 2.6e+18)
		tmp = t_1;
	elseif (u <= 5.6e+152)
		tmp = t1 / (-u * (u / v));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -3.7e-32], t$95$2, If[LessEqual[u, 5.2e-73], t$95$1, If[LessEqual[u, 1.8e-48], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.6e+18], t$95$1, If[LessEqual[u, 5.6e+152], N[(t1 / N[((-u) * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3.7e-32 or 5.6000000000000004e152 < u

   1. Initial program 75.4%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around 0 88.7%

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

      1. mul-1-neg 88.7%

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

      2. distribute-neg-frac 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in t1 around 0 86.7%

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

      1. frac-2neg 86.7%

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

      2. remove-double-neg 86.7%

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

      3. associate-*l/ 87.7%

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

   9. Applied egg-rr 87.7%

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

   if -3.7e-32 < u < 5.2000000000000002e-73 or 1.8000000000000001e-48 < u < 2.6e18

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 5.2000000000000002e-73 < u < 1.8000000000000001e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

      1. clear-num 86.5%

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

      2. frac-times 86.3%

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

      3. *-commutative 86.3%

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

      4. *-un-lft-identity 86.3%

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

      5. add-sqr-sqrt 54.7%

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

      6. sqrt-unprod 30.2%

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

      7. sqr-neg 30.2%

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

      8. sqrt-unprod 1.1%

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

      9. add-sqr-sqrt 13.3%

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

      10. frac-2neg 13.3%

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

      11. add-sqr-sqrt 1.2%

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

      12. sqrt-unprod 43.5%

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

      13. sqr-neg 43.5%

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

      14. sqrt-unprod 42.0%

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

      15. add-sqr-sqrt 86.3%

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

      16. distribute-neg-in 86.3%

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

      17. add-sqr-sqrt 55.2%

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

      18. sqrt-unprod 86.0%

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

      19. sqr-neg 86.0%

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

      20. sqrt-unprod 30.8%

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

      21. add-sqr-sqrt 86.0%

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

      22. sub-neg 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. associate-/r* 85.9%

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

      2. associate-/r/ 96.5%

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

   10. Simplified 96.5%

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

   if 2.6e18 < u < 5.6000000000000004e152

   1. Initial program 99.9%

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

      1. times-frac 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around 0 96.2%

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

      1. mul-1-neg 96.2%

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

      2. distribute-neg-frac 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in t1 around 0 76.2%

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

      1. *-commutative 76.2%

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

      2. clear-num 76.2%

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

      3. frac-2neg 76.2%

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

      4. frac-times 92.0%

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

      5. *-un-lft-identity 92.0%

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

      6. remove-double-neg 92.0%

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

   9. Applied egg-rr 92.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 7: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;u \leq -1.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-\frac{t1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= u -1.15e-32)
     (/ (/ t1 u) (/ (- t1 u) v))
     (if (<= u 8.2e-72)
       t_1
       (if (<= u 1.75e-48)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= u 1e+14) t_1 (* (/ v (+ t1 u)) (- (/ t1 u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (u <= -1.15e-32) {
		tmp = (t1 / u) / ((t1 - u) / v);
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 1.75e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 1e+14) {
		tmp = t_1;
	} else {
		tmp = (v / (t1 + 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) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    if (u <= (-1.15d-32)) then
        tmp = (t1 / u) / ((t1 - u) / v)
    else if (u <= 8.2d-72) then
        tmp = t_1
    else if (u <= 1.75d-48) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 1d+14) then
        tmp = t_1
    else
        tmp = (v / (t1 + u)) * -(t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (u <= -1.15e-32) {
		tmp = (t1 / u) / ((t1 - u) / v);
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 1.75e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 1e+14) {
		tmp = t_1;
	} else {
		tmp = (v / (t1 + u)) * -(t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	tmp = 0
	if u <= -1.15e-32:
		tmp = (t1 / u) / ((t1 - u) / v)
	elif u <= 8.2e-72:
		tmp = t_1
	elif u <= 1.75e-48:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 1e+14:
		tmp = t_1
	else:
		tmp = (v / (t1 + u)) * -(t1 / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (u <= -1.15e-32)
		tmp = Float64(Float64(t1 / u) / Float64(Float64(t1 - u) / v));
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 1.75e-48)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 1e+14)
		tmp = t_1;
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(-Float64(t1 / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (u <= -1.15e-32)
		tmp = (t1 / u) / ((t1 - u) / v);
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 1.75e-48)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 1e+14)
		tmp = t_1;
	else
		tmp = (v / (t1 + u)) * -(t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.15e-32], N[(N[(t1 / u), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.2e-72], t$95$1, If[LessEqual[u, 1.75e-48], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1e+14], t$95$1, N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * (-N[(t1 / u), $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.15e-32

   1. Initial program 77.2%

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

      1. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. distribute-neg-frac 85.1%

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

   6. Simplified 85.1%

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

      1. clear-num 85.9%

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

      2. un-div-inv 86.0%

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

      3. add-sqr-sqrt 55.6%

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

      4. sqrt-unprod 60.9%

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

      5. sqr-neg 60.9%

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

      6. sqrt-unprod 17.5%

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

      7. add-sqr-sqrt 48.0%

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

      8. frac-2neg 48.0%

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

      9. add-sqr-sqrt 25.4%

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

      10. sqrt-unprod 57.3%

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

      11. sqr-neg 57.3%

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

      12. sqrt-unprod 44.7%

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

      13. add-sqr-sqrt 86.0%

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

      14. distribute-neg-in 86.0%

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

      15. add-sqr-sqrt 55.7%

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

      16. sqrt-unprod 81.8%

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

      17. sqr-neg 81.8%

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

      18. sqrt-unprod 30.3%

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

      19. add-sqr-sqrt 85.6%

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

      20. sub-neg 85.6%

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

   8. Applied egg-rr 85.6%

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

   if -1.15e-32 < u < 8.20000000000000007e-72 or 1.74999999999999996e-48 < u < 1e14

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 8.20000000000000007e-72 < u < 1.74999999999999996e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

      1. expm1-log1p-u 55.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

   8. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 55.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)\right)} \]

      2. expm1-log1p 96.6%

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

   10. Simplified 96.6%

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

   if 1e14 < u

   1. Initial program 83.5%

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

      1. times-frac 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in t1 around 0 96.3%

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

      1. mul-1-neg 96.3%

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

      2. distribute-neg-frac 96.3%

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

   6. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{t1 + u} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 10^{+14}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-\frac{t1}{u}\right)\\ \end{array} \]

Alternative 8: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \mathbf{if}\;u \leq -7.4 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))) (t_2 (/ (/ t1 u) (/ (- t1 u) v))))
   (if (<= u -7.4e-35)
     t_2
     (if (<= u 8.2e-72)
       t_1
       (if (<= u 3.5e-48)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= u 2.15e+14) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 / u) / ((t1 - u) / v);
	double tmp;
	if (u <= -7.4e-35) {
		tmp = t_2;
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 3.5e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 2.15e+14) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = (t1 / u) / ((t1 - u) / v)
    if (u <= (-7.4d-35)) then
        tmp = t_2
    else if (u <= 8.2d-72) then
        tmp = t_1
    else if (u <= 3.5d-48) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (u <= 2.15d+14) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = (t1 / u) / ((t1 - u) / v);
	double tmp;
	if (u <= -7.4e-35) {
		tmp = t_2;
	} else if (u <= 8.2e-72) {
		tmp = t_1;
	} else if (u <= 3.5e-48) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (u <= 2.15e+14) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = (t1 / u) / ((t1 - u) / v)
	tmp = 0
	if u <= -7.4e-35:
		tmp = t_2
	elif u <= 8.2e-72:
		tmp = t_1
	elif u <= 3.5e-48:
		tmp = v / ((u / t1) * (t1 - u))
	elif u <= 2.15e+14:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(Float64(t1 / u) / Float64(Float64(t1 - u) / v))
	tmp = 0.0
	if (u <= -7.4e-35)
		tmp = t_2;
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 3.5e-48)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (u <= 2.15e+14)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = (t1 / u) / ((t1 - u) / v);
	tmp = 0.0;
	if (u <= -7.4e-35)
		tmp = t_2;
	elseif (u <= 8.2e-72)
		tmp = t_1;
	elseif (u <= 3.5e-48)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (u <= 2.15e+14)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 / u), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -7.4e-35], t$95$2, If[LessEqual[u, 8.2e-72], t$95$1, If[LessEqual[u, 3.5e-48], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.15e+14], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.3999999999999998e-35 or 2.15e14 < u

   1. Initial program 80.0%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in t1 around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. distribute-neg-frac 90.1%

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

   6. Simplified 90.1%

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

      1. clear-num 90.6%

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

      2. un-div-inv 90.6%

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

      3. add-sqr-sqrt 51.0%

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

      4. sqrt-unprod 61.3%

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

      5. sqr-neg 61.3%

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

      6. sqrt-unprod 26.9%

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

      7. add-sqr-sqrt 59.6%

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

      8. frac-2neg 59.6%

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

      9. add-sqr-sqrt 28.1%

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

      10. sqrt-unprod 66.0%

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

      11. sqr-neg 66.0%

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

      12. sqrt-unprod 48.2%

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

      13. add-sqr-sqrt 90.6%

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

      14. distribute-neg-in 90.6%

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

      15. add-sqr-sqrt 51.1%

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

      16. sqrt-unprod 86.2%

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

      17. sqr-neg 86.2%

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

      18. sqrt-unprod 39.5%

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

      19. add-sqr-sqrt 90.4%

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

      20. sub-neg 90.4%

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

   8. Applied egg-rr 90.4%

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

   if -7.3999999999999998e-35 < u < 8.20000000000000007e-72 or 3.49999999999999991e-48 < u < 2.15e14

   1. Initial program 65.5%

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

      1. times-frac 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-*l/ 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-frac-neg 97.6%

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

      7. +-commutative 97.6%

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

      8. remove-double-neg 97.6%

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

      9. unsub-neg 97.6%

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

      10. div-sub 97.6%

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

      11. sub-neg 97.6%

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

      12. distribute-frac-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. *-inverses 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 82.9%

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

      1. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   if 8.20000000000000007e-72 < u < 3.49999999999999991e-48

   1. Initial program 78.8%

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

      1. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-neg-frac 86.5%

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

   6. Simplified 86.5%

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

      1. expm1-log1p-u 55.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

   8. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 55.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\right)\right)} \]

      2. expm1-log1p 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1 - u}{v}}\\ \end{array} \]

Alternative 9: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{-34} \lor \neg \left(u \leq 5.4 \cdot 10^{-73} \lor \neg \left(u \leq 2.7 \cdot 10^{-48}\right) \land u \leq 1.12 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.6e-34)
         (not (or (<= u 5.4e-73) (and (not (<= u 2.7e-48)) (<= u 1.12e+55)))))
   (* (/ t1 u) (/ (- v) u))
   (/ (- v) (+ t1 (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.6e-34) || !((u <= 5.4e-73) || (!(u <= 2.7e-48) && (u <= 1.12e+55)))) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	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 ((u <= (-8.6d-34)) .or. (.not. (u <= 5.4d-73) .or. (.not. (u <= 2.7d-48)) .and. (u <= 1.12d+55))) then
        tmp = (t1 / u) * (-v / u)
    else
        tmp = -v / (t1 + (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.6e-34) || !((u <= 5.4e-73) || (!(u <= 2.7e-48) && (u <= 1.12e+55)))) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.6e-34) or not ((u <= 5.4e-73) or (not (u <= 2.7e-48) and (u <= 1.12e+55))):
		tmp = (t1 / u) * (-v / u)
	else:
		tmp = -v / (t1 + (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.6e-34) || !((u <= 5.4e-73) || (!(u <= 2.7e-48) && (u <= 1.12e+55))))
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.6e-34) || ~(((u <= 5.4e-73) || (~((u <= 2.7e-48)) && (u <= 1.12e+55)))))
		tmp = (t1 / u) * (-v / u);
	else
		tmp = -v / (t1 + (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.6e-34], N[Not[Or[LessEqual[u, 5.4e-73], And[N[Not[LessEqual[u, 2.7e-48]], $MachinePrecision], LessEqual[u, 1.12e+55]]]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.5999999999999999e-34 or 5.39999999999999989e-73 < u < 2.70000000000000011e-48 or 1.12000000000000006e55 < u

   1. Initial program 78.9%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in t1 around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac 90.0%

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

   6. Simplified 90.0%

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

   7. Taylor expanded in t1 around 0 86.9%

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

   if -8.5999999999999999e-34 < u < 5.39999999999999989e-73 or 2.70000000000000011e-48 < u < 1.12000000000000006e55

   1. Initial program 67.4%

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

      1. times-frac 96.9%

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

      2. neg-mul-1 96.9%

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

      3. associate-/l* 96.9%

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

      4. associate-*l/ 97.0%

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

      5. neg-mul-1 97.0%

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

      6. distribute-frac-neg 97.0%

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

      7. +-commutative 97.0%

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

      8. remove-double-neg 97.0%

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

      9. unsub-neg 97.0%

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

      10. div-sub 97.0%

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

      11. sub-neg 97.0%

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

      12. distribute-frac-neg 97.0%

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

      13. remove-double-neg 97.0%

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

      14. *-inverses 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in v around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-neg-frac 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in t1 around inf 81.5%

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

      1. *-commutative 81.5%

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

   9. Simplified 81.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{-34} \lor \neg \left(u \leq 5.4 \cdot 10^{-73} \lor \neg \left(u \leq 2.7 \cdot 10^{-48}\right) \land u \leq 1.12 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]

Alternative 10: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.5 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{-48} \lor \neg \left(u \leq 1.4 \cdot 10^{+60}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 u) (/ (- v) u))))
   (if (<= u -3.5e-32)
     t_1
     (if (<= u 8.2e-72)
       (/ (- v) t1)
       (if (or (<= u 3.5e-48) (not (<= u 1.4e+60))) t_1 (/ (- v) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / u) * (-v / u);
	double tmp;
	if (u <= -3.5e-32) {
		tmp = t_1;
	} else if (u <= 8.2e-72) {
		tmp = -v / t1;
	} else if ((u <= 3.5e-48) || !(u <= 1.4e+60)) {
		tmp = t_1;
	} else {
		tmp = -v / (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) :: t_1
    real(8) :: tmp
    t_1 = (t1 / u) * (-v / u)
    if (u <= (-3.5d-32)) then
        tmp = t_1
    else if (u <= 8.2d-72) then
        tmp = -v / t1
    else if ((u <= 3.5d-48) .or. (.not. (u <= 1.4d+60))) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 / u) * (-v / u);
	double tmp;
	if (u <= -3.5e-32) {
		tmp = t_1;
	} else if (u <= 8.2e-72) {
		tmp = -v / t1;
	} else if ((u <= 3.5e-48) || !(u <= 1.4e+60)) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / u) * (-v / u)
	tmp = 0
	if u <= -3.5e-32:
		tmp = t_1
	elif u <= 8.2e-72:
		tmp = -v / t1
	elif (u <= 3.5e-48) or not (u <= 1.4e+60):
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / u) * Float64(Float64(-v) / u))
	tmp = 0.0
	if (u <= -3.5e-32)
		tmp = t_1;
	elseif (u <= 8.2e-72)
		tmp = Float64(Float64(-v) / t1);
	elseif ((u <= 3.5e-48) || !(u <= 1.4e+60))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / u) * (-v / u);
	tmp = 0.0;
	if (u <= -3.5e-32)
		tmp = t_1;
	elseif (u <= 8.2e-72)
		tmp = -v / t1;
	elseif ((u <= 3.5e-48) || ~((u <= 1.4e+60)))
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.5e-32], t$95$1, If[LessEqual[u, 8.2e-72], N[((-v) / t1), $MachinePrecision], If[Or[LessEqual[u, 3.5e-48], N[Not[LessEqual[u, 1.4e+60]], $MachinePrecision]], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.4999999999999999e-32 or 8.20000000000000007e-72 < u < 3.49999999999999991e-48 or 1.4e60 < u

   1. Initial program 78.9%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in t1 around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac 90.0%

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

   6. Simplified 90.0%

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

   7. Taylor expanded in t1 around 0 86.9%

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

   if -3.4999999999999999e-32 < u < 8.20000000000000007e-72

   1. Initial program 64.4%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. neg-mul-1 83.7%

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

   6. Simplified 83.7%

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

   if 3.49999999999999991e-48 < u < 1.4e60

   1. Initial program 84.3%

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

      1. times-frac 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in t1 around inf 67.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{-48} \lor \neg \left(u \leq 1.4 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 11: 97.9% 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 73.2%

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

   1. times-frac 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 12: 67.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+141} \lor \neg \left(u \leq 5.5 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.7e+141) (not (<= u 5.5e+92)))
   (/ v (* u (/ u t1)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+141) || !(u <= 5.5e+92)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / (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 ((u <= (-2.7d+141)) .or. (.not. (u <= 5.5d+92))) then
        tmp = v / (u * (u / t1))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+141) || !(u <= 5.5e+92)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.7e+141) or not (u <= 5.5e+92):
		tmp = v / (u * (u / t1))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.7e+141) || !(u <= 5.5e+92))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.7e+141) || ~((u <= 5.5e+92)))
		tmp = v / (u * (u / t1));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.7e+141], N[Not[LessEqual[u, 5.5e+92]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.7000000000000001e141 or 5.50000000000000053e92 < u

   1. Initial program 74.4%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around 0 94.7%

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

      1. mul-1-neg 94.7%

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

      2. distribute-neg-frac 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in t1 around 0 93.5%

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

      1. clear-num 93.4%

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

      2. frac-times 79.8%

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

      3. *-un-lft-identity 79.8%

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

      4. add-sqr-sqrt 48.0%

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

      5. sqrt-unprod 63.8%

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

      6. sqr-neg 63.8%

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

      7. sqrt-unprod 26.8%

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

      8. add-sqr-sqrt 66.4%

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

   9. Applied egg-rr 66.4%

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

   if -2.7000000000000001e141 < u < 5.50000000000000053e92

   1. Initial program 72.7%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 67.9%

      \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+141} \lor \neg \left(u \leq 5.5 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 13: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1e+139)
   (/ v (* u (/ u t1)))
   (if (<= u 7.2e+60) (/ (- v) (+ t1 u)) (/ (* t1 (/ v u)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+139) {
		tmp = v / (u * (u / t1));
	} else if (u <= 7.2e+60) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 * (v / u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1d+139)) then
        tmp = v / (u * (u / t1))
    else if (u <= 7.2d+60) then
        tmp = -v / (t1 + u)
    else
        tmp = (t1 * (v / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+139) {
		tmp = v / (u * (u / t1));
	} else if (u <= 7.2e+60) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 * (v / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1e+139:
		tmp = v / (u * (u / t1))
	elif u <= 7.2e+60:
		tmp = -v / (t1 + u)
	else:
		tmp = (t1 * (v / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1e+139)
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	elseif (u <= 7.2e+60)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1e+139)
		tmp = v / (u * (u / t1));
	elseif (u <= 7.2e+60)
		tmp = -v / (t1 + u);
	else
		tmp = (t1 * (v / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1e+139], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.2e+60], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.00000000000000003e139

   1. Initial program 67.2%

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

      1. times-frac 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around 0 90.9%

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

      1. mul-1-neg 90.9%

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

      2. distribute-neg-frac 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in t1 around 0 90.9%

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

      1. clear-num 90.8%

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

      2. frac-times 73.9%

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

      3. *-un-lft-identity 73.9%

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

      4. add-sqr-sqrt 54.8%

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

      5. sqrt-unprod 64.5%

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

      6. sqr-neg 64.5%

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

      7. sqrt-unprod 18.9%

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

      8. add-sqr-sqrt 61.7%

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

   9. Applied egg-rr 61.7%

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

   if -1.00000000000000003e139 < u < 7.19999999999999935e60

   1. Initial program 72.1%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around inf 67.8%

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

   if 7.19999999999999935e60 < u

   1. Initial program 81.2%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 97.6%

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

      1. mul-1-neg 97.6%

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

      2. distribute-neg-frac 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in t1 around 0 93.7%

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

      1. associate-*l/ 95.5%

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

      2. add-sqr-sqrt 49.5%

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

      3. sqrt-unprod 64.1%

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

      4. sqr-neg 64.1%

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

      5. sqrt-unprod 33.8%

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

      6. add-sqr-sqrt 72.3%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \]

Alternative 14: 56.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.32 \cdot 10^{+67} \lor \neg \left(u \leq 1.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.32e+67) (not (<= u 1.8e+61))) (/ -0.5 (/ u v)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.32e+67) || !(u <= 1.8e+61)) {
		tmp = -0.5 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.32d+67)) .or. (.not. (u <= 1.8d+61))) then
        tmp = (-0.5d0) / (u / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.32e+67) || !(u <= 1.8e+61)) {
		tmp = -0.5 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.32e+67) or not (u <= 1.8e+61):
		tmp = -0.5 / (u / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.32e+67) || !(u <= 1.8e+61))
		tmp = Float64(-0.5 / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.32e+67) || ~((u <= 1.8e+61)))
		tmp = -0.5 / (u / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.32e+67], N[Not[LessEqual[u, 1.8e+61]], $MachinePrecision]], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.3200000000000001e67 or 1.80000000000000005e61 < u

   1. Initial program 75.3%

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

      1. times-frac 98.8%

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

      2. neg-mul-1 98.8%

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

      3. associate-/l* 98.7%

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

      4. associate-*l/ 98.8%

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

      5. neg-mul-1 98.8%

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

      6. distribute-frac-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. remove-double-neg 98.8%

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

      9. unsub-neg 98.8%

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

      10. div-sub 98.8%

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

      11. sub-neg 98.8%

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

      12. distribute-frac-neg 98.8%

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

      13. remove-double-neg 98.8%

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

      14. *-inverses 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in v around 0 88.0%

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

      1. mul-1-neg 88.0%

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

      2. +-commutative 88.0%

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

      3. *-commutative 88.0%

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

      4. distribute-neg-frac 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in t1 around inf 45.0%

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

      1. *-commutative 45.0%

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

   9. Simplified 45.0%

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

   10. Taylor expanded in t1 around 0 36.4%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 36.4%

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

       2. associate-/l* 37.0%

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

   12. Simplified 37.0%

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

   if -1.3200000000000001e67 < u < 1.80000000000000005e61

   1. Initial program 71.9%

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

      1. times-frac 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in t1 around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   6. Simplified 68.7%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.32 \cdot 10^{+67} \lor \neg \left(u \leq 1.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 15: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+133} \lor \neg \left(u \leq 4.8 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.4e+133) (not (<= u 4.8e+99))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.4e+133) || !(u <= 4.8e+99)) {
		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 <= (-6.4d+133)) .or. (.not. (u <= 4.8d+99))) 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 <= -6.4e+133) || !(u <= 4.8e+99)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.4e+133) or not (u <= 4.8e+99):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.4e+133) || !(u <= 4.8e+99))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.4e+133) || ~((u <= 4.8e+99)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.4e+133], N[Not[LessEqual[u, 4.8e+99]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.39999999999999994e133 or 4.8000000000000002e99 < u

   1. Initial program 73.8%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around 0 94.6%

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

      1. mul-1-neg 94.6%

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

      2. distribute-neg-frac 94.6%

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

   6. Simplified 94.6%

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

      1. *-commutative 94.6%

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

      2. clear-num 94.5%

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

      3. frac-times 87.5%

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

      4. *-un-lft-identity 87.5%

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

      5. add-sqr-sqrt 50.6%

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

      6. sqrt-unprod 65.2%

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

      7. sqr-neg 65.2%

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

      8. sqrt-unprod 30.0%

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

      9. add-sqr-sqrt 69.5%

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

      10. frac-2neg 69.5%

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

      11. add-sqr-sqrt 34.0%

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

      12. sqrt-unprod 70.3%

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

      13. sqr-neg 70.3%

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

      14. sqrt-unprod 49.1%

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

      15. add-sqr-sqrt 87.5%

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

      16. distribute-neg-in 87.5%

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

      17. add-sqr-sqrt 50.6%

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

      18. sqrt-unprod 84.1%

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

      19. sqr-neg 84.1%

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

      20. sqrt-unprod 37.0%

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

      21. add-sqr-sqrt 87.6%

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

      22. sub-neg 87.6%

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

   8. Applied egg-rr 87.6%

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

   9. Taylor expanded in t1 around inf 37.6%

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

   if -6.39999999999999994e133 < u < 4.8000000000000002e99

   1. Initial program 73.0%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   6. Simplified 64.3%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+133} \lor \neg \left(u \leq 4.8 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 16: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+87}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.15e+87) (/ (- v) u) (if (<= u 5e+99) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.15e+87) {
		tmp = -v / u;
	} else if (u <= 5e+99) {
		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 <= (-2.15d+87)) then
        tmp = -v / u
    else if (u <= 5d+99) 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 <= -2.15e+87) {
		tmp = -v / u;
	} else if (u <= 5e+99) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.15e+87:
		tmp = -v / u
	elif u <= 5e+99:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.15e+87)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 5e+99)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.15e+87)
		tmp = -v / u;
	elseif (u <= 5e+99)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.15e+87], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 5e+99], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.15e87

   1. Initial program 69.4%

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

      1. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around 0 88.3%

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

      1. mul-1-neg 88.3%

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

      2. distribute-neg-frac 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in t1 around inf 37.0%

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

      1. associate-*r/ 37.0%

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

      2. neg-mul-1 37.0%

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

   9. Simplified 37.0%

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

   if -2.15e87 < u < 5.00000000000000008e99

   1. Initial program 73.0%

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

      1. times-frac 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t1 around inf 66.5%

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

      1. associate-*r/ 66.5%

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

      2. neg-mul-1 66.5%

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

   6. Simplified 66.5%

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

   if 5.00000000000000008e99 < u

   1. Initial program 78.2%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t1 around 0 97.3%

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

      1. mul-1-neg 97.3%

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

      2. distribute-neg-frac 97.3%

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

   6. Simplified 97.3%

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

      1. *-commutative 97.3%

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

      2. clear-num 97.2%

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

      3. frac-times 88.9%

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

      4. *-un-lft-identity 88.9%

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

      5. add-sqr-sqrt 46.2%

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

      6. sqrt-unprod 63.1%

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

      7. sqr-neg 63.1%

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

      8. sqrt-unprod 36.9%

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

      9. add-sqr-sqrt 74.7%

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

      10. frac-2neg 74.7%

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

      11. add-sqr-sqrt 34.8%

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

      12. sqrt-unprod 74.0%

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

      13. sqr-neg 74.0%

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

      14. sqrt-unprod 48.3%

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

      15. add-sqr-sqrt 88.9%

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

      16. distribute-neg-in 88.9%

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

      17. add-sqr-sqrt 46.2%

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

      18. sqrt-unprod 85.0%

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

      19. sqr-neg 85.0%

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

      20. sqrt-unprod 42.7%

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

      21. add-sqr-sqrt 88.9%

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

      22. sub-neg 88.9%

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

   8. Applied egg-rr 88.9%

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

   9. Taylor expanded in t1 around inf 37.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+87}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 17: 23.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+124} \lor \neg \left(t1 \leq 1.08 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.5e+124) (not (<= t1 1.08e+36))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e+124) || !(t1 <= 1.08e+36)) {
		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 <= (-9.5d+124)) .or. (.not. (t1 <= 1.08d+36))) 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 <= -9.5e+124) || !(t1 <= 1.08e+36)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.5e+124) or not (t1 <= 1.08e+36):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.5e+124) || !(t1 <= 1.08e+36))
		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 <= -9.5e+124) || ~((t1 <= 1.08e+36)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.5e+124], N[Not[LessEqual[t1, 1.08e+36]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.50000000000000004e124 or 1.08000000000000001e36 < t1

   1. Initial program 56.4%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. clear-num 99.1%

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

      3. frac-times 76.6%

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

      4. remove-double-neg 76.6%

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

      5. *-commutative 76.6%

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

      6. *-un-lft-identity 76.6%

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

      7. distribute-neg-in 76.6%

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

      8. add-sqr-sqrt 30.6%

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

      9. sqrt-unprod 52.9%

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

      10. sqr-neg 52.9%

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

      11. sqrt-unprod 32.3%

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

      12. add-sqr-sqrt 52.4%

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

      13. sub-neg 52.4%

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

   5. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in t1 around inf 38.3%

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

   if -9.50000000000000004e124 < t1 < 1.08000000000000001e36

   1. Initial program 83.1%

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

      1. times-frac 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around 0 68.6%

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

      1. mul-1-neg 68.6%

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

      2. distribute-neg-frac 68.6%

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

   6. Simplified 68.6%

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

      1. *-commutative 68.6%

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

      2. clear-num 68.6%

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

      3. frac-times 69.1%

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

      4. *-un-lft-identity 69.1%

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

      5. add-sqr-sqrt 40.9%

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

      6. sqrt-unprod 48.2%

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

      7. sqr-neg 48.2%

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

      8. sqrt-unprod 15.6%

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

      9. add-sqr-sqrt 36.6%

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

      10. frac-2neg 36.6%

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

      11. add-sqr-sqrt 18.9%

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

      12. sqrt-unprod 44.5%

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

      13. sqr-neg 44.5%

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

      14. sqrt-unprod 35.0%

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

      15. add-sqr-sqrt 69.1%

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

      16. distribute-neg-in 69.1%

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

      17. add-sqr-sqrt 41.0%

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

      18. sqrt-unprod 68.5%

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

      19. sqr-neg 68.5%

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

      20. sqrt-unprod 27.4%

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

      21. add-sqr-sqrt 68.2%

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

      22. sub-neg 68.2%

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

   8. Applied egg-rr 68.2%

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

   9. Taylor expanded in t1 around inf 17.4%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+124} \lor \neg \left(t1 \leq 1.08 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 18: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (- v) (+ t1 u)))

C

double code(double u, double v, double t1) {
	return -v / (t1 + u);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = -v / (t1 + u)
end function

Java

public static double code(double u, double v, double t1) {
	return -v / (t1 + u);
}

Python

def code(u, v, t1):
	return -v / (t1 + u)

Julia

function code(u, v, t1)
	return Float64(Float64(-v) / Float64(t1 + u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -v / (t1 + u);
end

Wolfram

code[u_, v_, t1_] := N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 97.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

4. Taylor expanded in t1 around inf 59.5%

   \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]

5. Final simplification 59.5%

   \[\leadsto \frac{-v}{t1 + u} \]

Alternative 19: 13.5% 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 73.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 97.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
4. Step-by-step derivation

   1. frac-2neg 97.6%

      \[\leadsto \color{blue}{\frac{-\left(-t1\right)}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   2. clear-num 97.3%

      \[\leadsto \frac{-\left(-t1\right)}{-\left(t1 + u\right)} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

   3. frac-times 84.9%

      \[\leadsto \color{blue}{\frac{\left(-\left(-t1\right)\right) \cdot 1}{\left(-\left(t1 + u\right)\right) \cdot \frac{t1 + u}{v}}} \]

   4. remove-double-neg 84.9%

      \[\leadsto \frac{\color{blue}{t1} \cdot 1}{\left(-\left(t1 + u\right)\right) \cdot \frac{t1 + u}{v}} \]

   5. *-commutative 84.9%

      \[\leadsto \frac{\color{blue}{1 \cdot t1}}{\left(-\left(t1 + u\right)\right) \cdot \frac{t1 + u}{v}} \]

   6. *-un-lft-identity 84.9%

      \[\leadsto \frac{\color{blue}{t1}}{\left(-\left(t1 + u\right)\right) \cdot \frac{t1 + u}{v}} \]

   7. distribute-neg-in 84.9%

      \[\leadsto \frac{t1}{\color{blue}{\left(\left(-t1\right) + \left(-u\right)\right)} \cdot \frac{t1 + u}{v}} \]

   8. add-sqr-sqrt 43.6%

      \[\leadsto \frac{t1}{\left(\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)\right) \cdot \frac{t1 + u}{v}} \]

   9. sqrt-unprod 68.1%

      \[\leadsto \frac{t1}{\left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)\right) \cdot \frac{t1 + u}{v}} \]

   10. sqr-neg 68.1%

       \[\leadsto \frac{t1}{\left(\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)\right) \cdot \frac{t1 + u}{v}} \]

   11. sqrt-unprod 28.7%

       \[\leadsto \frac{t1}{\left(\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)\right) \cdot \frac{t1 + u}{v}} \]

   12. add-sqr-sqrt 61.8%

       \[\leadsto \frac{t1}{\left(\color{blue}{t1} + \left(-u\right)\right) \cdot \frac{t1 + u}{v}} \]

   13. sub-neg 61.8%

       \[\leadsto \frac{t1}{\color{blue}{\left(t1 - u\right)} \cdot \frac{t1 + u}{v}} \]

5. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\frac{t1}{\left(t1 - u\right) \cdot \frac{t1 + u}{v}}} \]
6. Step-by-step derivation

   1. *-commutative 61.8%

      \[\leadsto \frac{t1}{\color{blue}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}} \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}} \]

8. Taylor expanded in t1 around inf 17.1%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]

9. Final simplification 17.1%

   \[\leadsto \frac{v}{t1} \]

Reproduce

herbie shell --seed 2023320 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))