Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.0%

Time: 13.0s

Alternatives: 18

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: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{t1 + u}}{\mathsf{fma}\left(-1, \frac{u}{t1}, -1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. associate-/r* 87.6%

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

   2. *-commutative 87.6%

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

   3. associate-/l* 98.6%

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

   4. associate-/l/ 95.4%

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

   5. +-commutative 95.4%

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

   6. remove-double-neg 95.4%

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

   7. unsub-neg 95.4%

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

   8. div-sub 95.4%

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

   9. sub-neg 95.4%

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

   10. *-inverses 95.4%

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

   11. metadata-eval 95.4%

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

3. Simplified 95.4%

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

4. Taylor expanded in v around 0 95.4%

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

   1. associate-/r* 97.7%

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

   2. fma-neg 97.7%

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

   3. metadata-eval 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 2: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{if}\;u \leq -1.05 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -8.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot -2}\\ \mathbf{elif}\;u \leq -1.66 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v u) (/ t1 (- t1 u)))))
   (if (<= u -1.05e+69)
     t_1
     (if (<= u -8.5e-33)
       (/ (- v) (+ t1 (* u -2.0)))
       (if (<= u -1.66e-56)
         t_1
         (if (<= u 7.6e-88)
           (/ (- v) t1)
           (if (<= u 2.5e-31)
             (* v (/ (/ t1 u) (- t1 u)))
             (if (<= u 8.5e+58) (/ v (- (* u -2.0) t1)) t_1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -1.05e+69) {
		tmp = t_1;
	} else if (u <= -8.5e-33) {
		tmp = -v / (t1 + (u * -2.0));
	} else if (u <= -1.66e-56) {
		tmp = t_1;
	} else if (u <= 7.6e-88) {
		tmp = -v / t1;
	} else if (u <= 2.5e-31) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 8.5e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / u) * (t1 / (t1 - u))
    if (u <= (-1.05d+69)) then
        tmp = t_1
    else if (u <= (-8.5d-33)) then
        tmp = -v / (t1 + (u * (-2.0d0)))
    else if (u <= (-1.66d-56)) then
        tmp = t_1
    else if (u <= 7.6d-88) then
        tmp = -v / t1
    else if (u <= 2.5d-31) then
        tmp = v * ((t1 / u) / (t1 - u))
    else if (u <= 8.5d+58) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -1.05e+69) {
		tmp = t_1;
	} else if (u <= -8.5e-33) {
		tmp = -v / (t1 + (u * -2.0));
	} else if (u <= -1.66e-56) {
		tmp = t_1;
	} else if (u <= 7.6e-88) {
		tmp = -v / t1;
	} else if (u <= 2.5e-31) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 8.5e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) * (t1 / (t1 - u))
	tmp = 0
	if u <= -1.05e+69:
		tmp = t_1
	elif u <= -8.5e-33:
		tmp = -v / (t1 + (u * -2.0))
	elif u <= -1.66e-56:
		tmp = t_1
	elif u <= 7.6e-88:
		tmp = -v / t1
	elif u <= 2.5e-31:
		tmp = v * ((t1 / u) / (t1 - u))
	elif u <= 8.5e+58:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) * Float64(t1 / Float64(t1 - u)))
	tmp = 0.0
	if (u <= -1.05e+69)
		tmp = t_1;
	elseif (u <= -8.5e-33)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * -2.0)));
	elseif (u <= -1.66e-56)
		tmp = t_1;
	elseif (u <= 7.6e-88)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 2.5e-31)
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)));
	elseif (u <= 8.5e+58)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / u) * (t1 / (t1 - u));
	tmp = 0.0;
	if (u <= -1.05e+69)
		tmp = t_1;
	elseif (u <= -8.5e-33)
		tmp = -v / (t1 + (u * -2.0));
	elseif (u <= -1.66e-56)
		tmp = t_1;
	elseif (u <= 7.6e-88)
		tmp = -v / t1;
	elseif (u <= 2.5e-31)
		tmp = v * ((t1 / u) / (t1 - u));
	elseif (u <= 8.5e+58)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.05e+69], t$95$1, If[LessEqual[u, -8.5e-33], N[((-v) / N[(t1 + N[(u * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.66e-56], t$95$1, If[LessEqual[u, 7.6e-88], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 2.5e-31], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.5e+58], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1.05000000000000008e69 or -8.49999999999999945e-33 < u < -1.66000000000000001e-56 or 8.50000000000000015e58 < u

   1. Initial program 83.2%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.8%

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

      3. frac-2neg 98.8%

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

      4. frac-times 91.1%

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

      5. *-un-lft-identity 91.1%

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

      6. remove-double-neg 91.1%

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

      7. distribute-neg-in 91.1%

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

      8. add-sqr-sqrt 50.1%

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

      9. sqrt-unprod 86.5%

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

      10. sqr-neg 86.5%

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

      11. sqrt-unprod 38.1%

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

      12. add-sqr-sqrt 86.5%

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

      13. sub-neg 86.5%

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

   5. Applied egg-rr 86.5%

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

   6. Taylor expanded in t1 around 0 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. times-frac 89.9%

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

      3. clear-num 89.9%

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

   8. Applied egg-rr 89.9%

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

   if -1.05000000000000008e69 < u < -8.49999999999999945e-33

   1. Initial program 69.5%

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

      1. times-frac 94.9%

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

   3. Simplified 94.9%

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

      1. clear-num 94.7%

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

      2. frac-2neg 94.7%

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

      3. frac-times 94.5%

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

      4. *-un-lft-identity 94.5%

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

      5. frac-2neg 94.5%

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

      6. distribute-neg-in 94.5%

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

      7. add-sqr-sqrt 36.9%

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

      8. sqrt-unprod 64.5%

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

      9. sqr-neg 64.5%

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

      10. sqrt-unprod 37.2%

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

      11. add-sqr-sqrt 53.7%

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

      12. sub-neg 53.7%

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

      13. remove-double-neg 53.7%

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

      14. distribute-neg-in 53.7%

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

      15. add-sqr-sqrt 16.5%

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

      16. sqrt-unprod 59.3%

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

      17. sqr-neg 59.3%

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

      18. sqrt-unprod 57.4%

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

      19. add-sqr-sqrt 94.5%

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

      20. sub-neg 94.5%

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

   5. Applied egg-rr 94.5%

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

      1. associate-*l/ 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in t1 around inf 64.8%

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

      1. *-commutative 64.8%

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

   10. Simplified 64.8%

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

   if -1.66000000000000001e-56 < u < 7.60000000000000022e-88

   1. Initial program 67.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if 7.60000000000000022e-88 < u < 2.5e-31

   1. Initial program 89.4%

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

      1. *-commutative 89.2%

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

      2. clear-num 89.0%

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

      3. frac-2neg 89.0%

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

      4. frac-times 78.3%

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

      5. *-un-lft-identity 78.3%

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

      6. remove-double-neg 78.3%

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

      7. distribute-neg-in 78.3%

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

      8. add-sqr-sqrt 66.8%

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

      9. sqrt-unprod 67.6%

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

      10. sqr-neg 67.6%

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

      11. sqrt-unprod 0.9%

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

      12. add-sqr-sqrt 67.6%

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

      13. sub-neg 67.6%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in t1 around 0 67.6%

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

   7. Taylor expanded in v around 0 68.3%

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

      1. associate-*l/ 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/r* 78.6%

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

   9. Simplified 78.6%

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

   if 2.5e-31 < u < 8.50000000000000015e58

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{elif}\;u \leq -8.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot -2}\\ \mathbf{elif}\;u \leq -1.66 \cdot 10^{-56}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \]

Alternative 3: 76.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ t_2 := \frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{if}\;u \leq -4.2 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -6.9 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -9.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)) (t_2 (* (/ v u) (/ t1 (- t1 u)))))
   (if (<= u -4.2e+68)
     t_2
     (if (<= u -6.9e+59)
       t_1
       (if (<= u -9.7e-57)
         (/ t1 (* (- t1 u) (/ u v)))
         (if (<= u 7.6e-92)
           t_1
           (if (<= u 7.5e-32)
             (* v (/ (/ t1 u) (- t1 u)))
             (if (<= u 1.55e+60) (/ v (- (* u -2.0) t1)) t_2))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double t_2 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -4.2e+68) {
		tmp = t_2;
	} else if (u <= -6.9e+59) {
		tmp = t_1;
	} else if (u <= -9.7e-57) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else if (u <= 7.6e-92) {
		tmp = t_1;
	} else if (u <= 7.5e-32) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 1.55e+60) {
		tmp = v / ((u * -2.0) - t1);
	} 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
    t_2 = (v / u) * (t1 / (t1 - u))
    if (u <= (-4.2d+68)) then
        tmp = t_2
    else if (u <= (-6.9d+59)) then
        tmp = t_1
    else if (u <= (-9.7d-57)) then
        tmp = t1 / ((t1 - u) * (u / v))
    else if (u <= 7.6d-92) then
        tmp = t_1
    else if (u <= 7.5d-32) then
        tmp = v * ((t1 / u) / (t1 - u))
    else if (u <= 1.55d+60) then
        tmp = v / ((u * (-2.0d0)) - t1)
    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;
	double t_2 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -4.2e+68) {
		tmp = t_2;
	} else if (u <= -6.9e+59) {
		tmp = t_1;
	} else if (u <= -9.7e-57) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else if (u <= 7.6e-92) {
		tmp = t_1;
	} else if (u <= 7.5e-32) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 1.55e+60) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	t_2 = (v / u) * (t1 / (t1 - u))
	tmp = 0
	if u <= -4.2e+68:
		tmp = t_2
	elif u <= -6.9e+59:
		tmp = t_1
	elif u <= -9.7e-57:
		tmp = t1 / ((t1 - u) * (u / v))
	elif u <= 7.6e-92:
		tmp = t_1
	elif u <= 7.5e-32:
		tmp = v * ((t1 / u) / (t1 - u))
	elif u <= 1.55e+60:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	t_2 = Float64(Float64(v / u) * Float64(t1 / Float64(t1 - u)))
	tmp = 0.0
	if (u <= -4.2e+68)
		tmp = t_2;
	elseif (u <= -6.9e+59)
		tmp = t_1;
	elseif (u <= -9.7e-57)
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	elseif (u <= 7.6e-92)
		tmp = t_1;
	elseif (u <= 7.5e-32)
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)));
	elseif (u <= 1.55e+60)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	t_2 = (v / u) * (t1 / (t1 - u));
	tmp = 0.0;
	if (u <= -4.2e+68)
		tmp = t_2;
	elseif (u <= -6.9e+59)
		tmp = t_1;
	elseif (u <= -9.7e-57)
		tmp = t1 / ((t1 - u) * (u / v));
	elseif (u <= 7.6e-92)
		tmp = t_1;
	elseif (u <= 7.5e-32)
		tmp = v * ((t1 / u) / (t1 - u));
	elseif (u <= 1.55e+60)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.2e+68], t$95$2, If[LessEqual[u, -6.9e+59], t$95$1, If[LessEqual[u, -9.7e-57], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.6e-92], t$95$1, If[LessEqual[u, 7.5e-32], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.55e+60], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -4.20000000000000002e68 or 1.55e60 < u

   1. Initial program 83.7%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-2neg 98.8%

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

      4. frac-times 91.7%

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

      5. *-un-lft-identity 91.7%

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

      6. remove-double-neg 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. add-sqr-sqrt 50.1%

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

      9. sqrt-unprod 86.7%

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

      10. sqr-neg 86.7%

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

      11. sqrt-unprod 38.4%

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

      12. add-sqr-sqrt 87.6%

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

      13. sub-neg 87.6%

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

   5. Applied egg-rr 87.6%

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

   6. Taylor expanded in t1 around 0 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. times-frac 90.9%

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

      3. clear-num 90.9%

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

   8. Applied egg-rr 90.9%

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

   if -4.20000000000000002e68 < u < -6.8999999999999998e59 or -9.7e-57 < u < 7.6000000000000001e-92

   1. Initial program 65.6%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   6. Simplified 87.2%

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

   if -6.8999999999999998e59 < u < -9.7e-57

   1. Initial program 82.9%

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

      1. times-frac 95.9%

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

   3. Simplified 95.9%

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

      1. *-commutative 95.9%

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

      2. clear-num 95.7%

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

      3. frac-2neg 95.7%

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

      4. frac-times 81.8%

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

      5. *-un-lft-identity 81.8%

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

      6. remove-double-neg 81.8%

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

      7. distribute-neg-in 81.8%

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

      8. add-sqr-sqrt 37.8%

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

      9. sqrt-unprod 77.7%

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

      10. sqr-neg 77.7%

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

      11. sqrt-unprod 39.9%

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

      12. add-sqr-sqrt 69.4%

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

      13. sub-neg 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in t1 around 0 69.4%

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

   if 7.6000000000000001e-92 < u < 7.49999999999999953e-32

   1. Initial program 89.4%

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

      1. *-commutative 89.2%

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

      2. clear-num 89.0%

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

      3. frac-2neg 89.0%

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

      4. frac-times 78.3%

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

      5. *-un-lft-identity 78.3%

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

      6. remove-double-neg 78.3%

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

      7. distribute-neg-in 78.3%

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

      8. add-sqr-sqrt 66.8%

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

      9. sqrt-unprod 67.6%

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

      10. sqr-neg 67.6%

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

      11. sqrt-unprod 0.9%

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

      12. add-sqr-sqrt 67.6%

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

      13. sub-neg 67.6%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in t1 around 0 67.6%

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

   7. Taylor expanded in v around 0 68.3%

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

      1. associate-*l/ 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/r* 78.6%

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

   9. Simplified 78.6%

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

   if 7.49999999999999953e-32 < u < 1.55e60

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{elif}\;u \leq -6.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq -9.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \]

Alternative 4: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ t_2 := \frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{if}\;u \leq -4.2 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -6.9 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -6.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot \left(t1 - u\right)}{v}}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 7 \cdot 10^{+59}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)) (t_2 (* (/ v u) (/ t1 (- t1 u)))))
   (if (<= u -4.2e+68)
     t_2
     (if (<= u -6.9e+59)
       t_1
       (if (<= u -6.1e-56)
         (/ t1 (/ (* u (- t1 u)) v))
         (if (<= u 7.6e-88)
           t_1
           (if (<= u 2.1e-32)
             (* v (/ (/ t1 u) (- t1 u)))
             (if (<= u 7e+59) (/ v (- (* u -2.0) t1)) t_2))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double t_2 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -4.2e+68) {
		tmp = t_2;
	} else if (u <= -6.9e+59) {
		tmp = t_1;
	} else if (u <= -6.1e-56) {
		tmp = t1 / ((u * (t1 - u)) / v);
	} else if (u <= 7.6e-88) {
		tmp = t_1;
	} else if (u <= 2.1e-32) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 7e+59) {
		tmp = v / ((u * -2.0) - t1);
	} 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
    t_2 = (v / u) * (t1 / (t1 - u))
    if (u <= (-4.2d+68)) then
        tmp = t_2
    else if (u <= (-6.9d+59)) then
        tmp = t_1
    else if (u <= (-6.1d-56)) then
        tmp = t1 / ((u * (t1 - u)) / v)
    else if (u <= 7.6d-88) then
        tmp = t_1
    else if (u <= 2.1d-32) then
        tmp = v * ((t1 / u) / (t1 - u))
    else if (u <= 7d+59) then
        tmp = v / ((u * (-2.0d0)) - t1)
    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;
	double t_2 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -4.2e+68) {
		tmp = t_2;
	} else if (u <= -6.9e+59) {
		tmp = t_1;
	} else if (u <= -6.1e-56) {
		tmp = t1 / ((u * (t1 - u)) / v);
	} else if (u <= 7.6e-88) {
		tmp = t_1;
	} else if (u <= 2.1e-32) {
		tmp = v * ((t1 / u) / (t1 - u));
	} else if (u <= 7e+59) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	t_2 = (v / u) * (t1 / (t1 - u))
	tmp = 0
	if u <= -4.2e+68:
		tmp = t_2
	elif u <= -6.9e+59:
		tmp = t_1
	elif u <= -6.1e-56:
		tmp = t1 / ((u * (t1 - u)) / v)
	elif u <= 7.6e-88:
		tmp = t_1
	elif u <= 2.1e-32:
		tmp = v * ((t1 / u) / (t1 - u))
	elif u <= 7e+59:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	t_2 = Float64(Float64(v / u) * Float64(t1 / Float64(t1 - u)))
	tmp = 0.0
	if (u <= -4.2e+68)
		tmp = t_2;
	elseif (u <= -6.9e+59)
		tmp = t_1;
	elseif (u <= -6.1e-56)
		tmp = Float64(t1 / Float64(Float64(u * Float64(t1 - u)) / v));
	elseif (u <= 7.6e-88)
		tmp = t_1;
	elseif (u <= 2.1e-32)
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)));
	elseif (u <= 7e+59)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	t_2 = (v / u) * (t1 / (t1 - u));
	tmp = 0.0;
	if (u <= -4.2e+68)
		tmp = t_2;
	elseif (u <= -6.9e+59)
		tmp = t_1;
	elseif (u <= -6.1e-56)
		tmp = t1 / ((u * (t1 - u)) / v);
	elseif (u <= 7.6e-88)
		tmp = t_1;
	elseif (u <= 2.1e-32)
		tmp = v * ((t1 / u) / (t1 - u));
	elseif (u <= 7e+59)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.2e+68], t$95$2, If[LessEqual[u, -6.9e+59], t$95$1, If[LessEqual[u, -6.1e-56], N[(t1 / N[(N[(u * N[(t1 - u), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.6e-88], t$95$1, If[LessEqual[u, 2.1e-32], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7e+59], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -4.20000000000000002e68 or 7e59 < u

   1. Initial program 83.7%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. frac-2neg 98.8%

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

      4. frac-times 91.7%

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

      5. *-un-lft-identity 91.7%

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

      6. remove-double-neg 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. add-sqr-sqrt 50.1%

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

      9. sqrt-unprod 86.7%

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

      10. sqr-neg 86.7%

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

      11. sqrt-unprod 38.4%

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

      12. add-sqr-sqrt 87.6%

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

      13. sub-neg 87.6%

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

   5. Applied egg-rr 87.6%

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

   6. Taylor expanded in t1 around 0 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. times-frac 90.9%

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

      3. clear-num 90.9%

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

   8. Applied egg-rr 90.9%

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

   if -4.20000000000000002e68 < u < -6.8999999999999998e59 or -6.0999999999999998e-56 < u < 7.60000000000000022e-88

   1. Initial program 65.6%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   6. Simplified 87.2%

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

   if -6.8999999999999998e59 < u < -6.0999999999999998e-56

   1. Initial program 82.9%

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

      1. times-frac 95.9%

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

   3. Simplified 95.9%

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

      1. *-commutative 95.9%

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

      2. clear-num 95.7%

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

      3. frac-2neg 95.7%

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

      4. frac-times 81.8%

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

      5. *-un-lft-identity 81.8%

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

      6. remove-double-neg 81.8%

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

      7. distribute-neg-in 81.8%

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

      8. add-sqr-sqrt 37.8%

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

      9. sqrt-unprod 77.7%

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

      10. sqr-neg 77.7%

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

      11. sqrt-unprod 39.9%

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

      12. add-sqr-sqrt 69.4%

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

      13. sub-neg 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in t1 around 0 69.4%

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

   7. Taylor expanded in v around 0 69.4%

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

   if 7.60000000000000022e-88 < u < 2.0999999999999999e-32

   1. Initial program 89.4%

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

      1. *-commutative 89.2%

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

      2. clear-num 89.0%

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

      3. frac-2neg 89.0%

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

      4. frac-times 78.3%

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

      5. *-un-lft-identity 78.3%

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

      6. remove-double-neg 78.3%

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

      7. distribute-neg-in 78.3%

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

      8. add-sqr-sqrt 66.8%

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

      9. sqrt-unprod 67.6%

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

      10. sqr-neg 67.6%

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

      11. sqrt-unprod 0.9%

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

      12. add-sqr-sqrt 67.6%

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

      13. sub-neg 67.6%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in t1 around 0 67.6%

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

   7. Taylor expanded in v around 0 68.3%

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

      1. associate-*l/ 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/r* 78.6%

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

   9. Simplified 78.6%

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

   if 2.0999999999999999e-32 < u < 7e59

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{elif}\;u \leq -6.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq -6.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot \left(t1 - u\right)}{v}}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 7 \cdot 10^{+59}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \]

Alternative 5: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -2.2 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-216}:\\ \;\;\;\;\frac{\frac{v \cdot t1}{t1 - u}}{u}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.4e+154)
     (/ v (- (* u -2.0) t1))
     (if (<= t1 -2.2e-242)
       t_1
       (if (<= t1 7e-216)
         (/ (/ (* v t1) (- t1 u)) u)
         (if (<= t1 6.5e+136) t_1 (/ (- v) (+ t1 (* u -2.0)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.4e+154) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= -2.2e-242) {
		tmp = t_1;
	} else if (t1 <= 7e-216) {
		tmp = ((v * t1) / (t1 - u)) / u;
	} else if (t1 <= 6.5e+136) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.4d+154)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (t1 <= (-2.2d-242)) then
        tmp = t_1
    else if (t1 <= 7d-216) then
        tmp = ((v * t1) / (t1 - u)) / u
    else if (t1 <= 6.5d+136) then
        tmp = t_1
    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 t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.4e+154) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= -2.2e-242) {
		tmp = t_1;
	} else if (t1 <= 7e-216) {
		tmp = ((v * t1) / (t1 - u)) / u;
	} else if (t1 <= 6.5e+136) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + (u * -2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.4e+154:
		tmp = v / ((u * -2.0) - t1)
	elif t1 <= -2.2e-242:
		tmp = t_1
	elif t1 <= 7e-216:
		tmp = ((v * t1) / (t1 - u)) / u
	elif t1 <= 6.5e+136:
		tmp = t_1
	else:
		tmp = -v / (t1 + (u * -2.0))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -1.4e+154)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (t1 <= -2.2e-242)
		tmp = t_1;
	elseif (t1 <= 7e-216)
		tmp = Float64(Float64(Float64(v * t1) / Float64(t1 - u)) / u);
	elseif (t1 <= 6.5e+136)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -1.4e+154)
		tmp = v / ((u * -2.0) - t1);
	elseif (t1 <= -2.2e-242)
		tmp = t_1;
	elseif (t1 <= 7e-216)
		tmp = ((v * t1) / (t1 - u)) / u;
	elseif (t1 <= 6.5e+136)
		tmp = t_1;
	else
		tmp = -v / (t1 + (u * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.4e+154], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.2e-242], t$95$1, If[LessEqual[t1, 7e-216], N[(N[(N[(v * t1), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[t1, 6.5e+136], t$95$1, N[((-v) / N[(t1 + N[(u * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.4e154

   1. Initial program 51.5%

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

      1. associate-/r* 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. *-inverses 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. unsub-neg 92.2%

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

      3. *-commutative 92.2%

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

   6. Simplified 92.2%

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

   if -1.4e154 < t1 < -2.20000000000000002e-242 or 6.99999999999999965e-216 < t1 < 6.4999999999999998e136

   1. Initial program 87.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 v around 0 87.4%

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

      1. mul-1-neg 87.4%

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

      2. associate-*l/ 93.2%

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

      3. *-commutative 93.2%

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

      4. distribute-rgt-neg-in 93.2%

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

   6. Simplified 93.2%

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

      1. unpow2 93.2%

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

   8. Applied egg-rr 93.2%

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

   if -2.20000000000000002e-242 < t1 < 6.99999999999999965e-216

   1. Initial program 77.9%

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

      1. times-frac 90.5%

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

   3. Simplified 90.5%

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

      1. *-commutative 90.5%

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

      2. clear-num 90.4%

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

      3. frac-2neg 90.4%

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

      4. frac-times 82.0%

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

      5. *-un-lft-identity 82.0%

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

      6. remove-double-neg 82.0%

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

      7. distribute-neg-in 82.0%

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

      8. add-sqr-sqrt 38.3%

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

      9. sqrt-unprod 76.7%

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

      10. sqr-neg 76.7%

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

      11. sqrt-unprod 41.0%

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

      12. add-sqr-sqrt 76.5%

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

      13. sub-neg 76.5%

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

   5. Applied egg-rr 76.5%

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

   6. Taylor expanded in t1 around 0 79.3%

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

      1. associate-*l/ 76.7%

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

      2. associate-/r/ 72.1%

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

   8. Applied egg-rr 72.1%

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

      1. associate-*l/ 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-/r* 88.3%

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

   10. Applied egg-rr 88.3%

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

   if 6.4999999999999998e136 < t1

   1. Initial program 45.7%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 97.0%

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

      4. *-un-lft-identity 97.0%

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

      5. frac-2neg 97.0%

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

      6. distribute-neg-in 97.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 43.7%

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

      9. sqr-neg 43.7%

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

      10. sqrt-unprod 45.8%

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

      11. add-sqr-sqrt 45.8%

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

      12. sub-neg 45.8%

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

      13. remove-double-neg 45.8%

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

      14. distribute-neg-in 45.8%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 49.5%

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

      17. sqr-neg 49.5%

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

      18. sqrt-unprod 96.7%

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

      19. add-sqr-sqrt 97.0%

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

      20. sub-neg 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. associate-*l/ 49.5%

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

   7. Simplified 49.5%

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

   8. Taylor expanded in t1 around inf 91.5%

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

      1. *-commutative 91.5%

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

   10. Simplified 91.5%

       \[\leadsto \frac{-v}{\color{blue}{t1 + u \cdot -2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -2.2 \cdot 10^{-242}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-216}:\\ \;\;\;\;\frac{\frac{v \cdot t1}{t1 - u}}{u}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+136}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot -2}\\ \end{array} \]

Alternative 6: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{if}\;u \leq -5.5 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 9.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (+ t1 u)) (/ (- t1) u))))
   (if (<= u -5.5e-58)
     t_1
     (if (<= u 7.6e-88)
       (/ (- v) t1)
       (if (<= u 4.7e-32)
         t_1
         (if (<= u 9.6e+58)
           (/ v (- (* u -2.0) t1))
           (/ 1.0 (/ (- t1 u) (* t1 (/ v u))))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) * (-t1 / u);
	double tmp;
	if (u <= -5.5e-58) {
		tmp = t_1;
	} else if (u <= 7.6e-88) {
		tmp = -v / t1;
	} else if (u <= 4.7e-32) {
		tmp = t_1;
	} else if (u <= 9.6e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 1.0 / ((t1 - u) / (t1 * (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) :: t_1
    real(8) :: tmp
    t_1 = (v / (t1 + u)) * (-t1 / u)
    if (u <= (-5.5d-58)) then
        tmp = t_1
    else if (u <= 7.6d-88) then
        tmp = -v / t1
    else if (u <= 4.7d-32) then
        tmp = t_1
    else if (u <= 9.6d+58) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = 1.0d0 / ((t1 - u) / (t1 * (v / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) * (-t1 / u);
	double tmp;
	if (u <= -5.5e-58) {
		tmp = t_1;
	} else if (u <= 7.6e-88) {
		tmp = -v / t1;
	} else if (u <= 4.7e-32) {
		tmp = t_1;
	} else if (u <= 9.6e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (t1 + u)) * (-t1 / u)
	tmp = 0
	if u <= -5.5e-58:
		tmp = t_1
	elif u <= 7.6e-88:
		tmp = -v / t1
	elif u <= 4.7e-32:
		tmp = t_1
	elif u <= 9.6e+58:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u))
	tmp = 0.0
	if (u <= -5.5e-58)
		tmp = t_1;
	elseif (u <= 7.6e-88)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 4.7e-32)
		tmp = t_1;
	elseif (u <= 9.6e+58)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(1.0 / Float64(Float64(t1 - u) / Float64(t1 * Float64(v / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / (t1 + u)) * (-t1 / u);
	tmp = 0.0;
	if (u <= -5.5e-58)
		tmp = t_1;
	elseif (u <= 7.6e-88)
		tmp = -v / t1;
	elseif (u <= 4.7e-32)
		tmp = t_1;
	elseif (u <= 9.6e+58)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.5e-58], t$95$1, If[LessEqual[u, 7.6e-88], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 4.7e-32], t$95$1, If[LessEqual[u, 9.6e+58], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t1 - u), $MachinePrecision] / N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.49999999999999996e-58 or 7.60000000000000022e-88 < u < 4.70000000000000019e-32

   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 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 0 82.2%

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

      1. associate-*r/ 82.2%

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

      2. mul-1-neg 82.2%

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

   6. Simplified 82.2%

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

   if -5.49999999999999996e-58 < u < 7.60000000000000022e-88

   1. Initial program 67.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if 4.70000000000000019e-32 < u < 9.5999999999999999e58

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   if 9.5999999999999999e58 < u

   1. Initial program 84.9%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. clear-num 98.0%

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

      3. frac-2neg 98.0%

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

      4. frac-times 94.2%

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

      5. *-un-lft-identity 94.2%

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

      6. remove-double-neg 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. add-sqr-sqrt 55.6%

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

      9. sqrt-unprod 90.4%

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

      10. sqr-neg 90.4%

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

      11. sqrt-unprod 34.8%

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

      12. add-sqr-sqrt 90.4%

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

      13. sub-neg 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in t1 around 0 90.4%

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

      1. clear-num 90.3%

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

      2. inv-pow 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-/l* 93.8%

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

      5. div-inv 93.8%

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

      6. clear-num 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. unpow-1 93.9%

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

   10. Simplified 93.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 9.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}\\ \end{array} \]

Alternative 7: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{-31}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8.2e-58)
   (* (/ v (+ t1 u)) (/ (- t1) u))
   (if (<= u 3.8e-88)
     (/ (- v) t1)
     (if (<= u 4e-31)
       (/ v (* (+ t1 u) (/ (- t1 u) t1)))
       (if (<= u 8.5e+58)
         (/ v (- (* u -2.0) t1))
         (/ 1.0 (/ (- t1 u) (* t1 (/ v u)))))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.2e-58) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= 3.8e-88) {
		tmp = -v / t1;
	} else if (u <= 4e-31) {
		tmp = v / ((t1 + u) * ((t1 - u) / t1));
	} else if (u <= 8.5e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 1.0 / ((t1 - u) / (t1 * (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 <= (-8.2d-58)) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else if (u <= 3.8d-88) then
        tmp = -v / t1
    else if (u <= 4d-31) then
        tmp = v / ((t1 + u) * ((t1 - u) / t1))
    else if (u <= 8.5d+58) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = 1.0d0 / ((t1 - u) / (t1 * (v / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.2e-58) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= 3.8e-88) {
		tmp = -v / t1;
	} else if (u <= 4e-31) {
		tmp = v / ((t1 + u) * ((t1 - u) / t1));
	} else if (u <= 8.5e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8.2e-58:
		tmp = (v / (t1 + u)) * (-t1 / u)
	elif u <= 3.8e-88:
		tmp = -v / t1
	elif u <= 4e-31:
		tmp = v / ((t1 + u) * ((t1 - u) / t1))
	elif u <= 8.5e+58:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8.2e-58)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	elseif (u <= 3.8e-88)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 4e-31)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(Float64(t1 - u) / t1)));
	elseif (u <= 8.5e+58)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(1.0 / Float64(Float64(t1 - u) / Float64(t1 * Float64(v / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8.2e-58)
		tmp = (v / (t1 + u)) * (-t1 / u);
	elseif (u <= 3.8e-88)
		tmp = -v / t1;
	elseif (u <= 4e-31)
		tmp = v / ((t1 + u) * ((t1 - u) / t1));
	elseif (u <= 8.5e+58)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8.2e-58], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.8e-88], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 4e-31], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.5e+58], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t1 - u), $MachinePrecision] / N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -8.20000000000000056e-58

   1. Initial program 79.0%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in t1 around 0 82.6%

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

      1. associate-*r/ 82.6%

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

      2. mul-1-neg 82.6%

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

   6. Simplified 82.6%

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

   if -8.20000000000000056e-58 < u < 3.80000000000000011e-88

   1. Initial program 67.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if 3.80000000000000011e-88 < u < 4e-31

   1. Initial program 89.4%

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

      1. clear-num 89.3%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. frac-2neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. add-sqr-sqrt 77.4%

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

      7. sqrt-unprod 88.7%

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

      8. sqr-neg 88.7%

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

      9. sqrt-unprod 11.3%

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

      10. add-sqr-sqrt 88.7%

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

      11. sub-neg 88.7%

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

      12. remove-double-neg 88.7%

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

   5. Applied egg-rr 88.7%

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

   if 4e-31 < u < 8.50000000000000015e58

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   if 8.50000000000000015e58 < u

   1. Initial program 84.9%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. clear-num 98.0%

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

      3. frac-2neg 98.0%

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

      4. frac-times 94.2%

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

      5. *-un-lft-identity 94.2%

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

      6. remove-double-neg 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. add-sqr-sqrt 55.6%

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

      9. sqrt-unprod 90.4%

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

      10. sqr-neg 90.4%

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

      11. sqrt-unprod 34.8%

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

      12. add-sqr-sqrt 90.4%

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

      13. sub-neg 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in t1 around 0 90.4%

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

      1. clear-num 90.3%

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

      2. inv-pow 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-/l* 93.8%

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

      5. div-inv 93.8%

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

      6. clear-num 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. unpow-1 93.9%

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

   10. Simplified 93.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{-31}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}\\ \end{array} \]

Alternative 8: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 - u}{t1}\\ \mathbf{if}\;u \leq -1.02 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{v}{t_1}}{t1 + u}\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot t_1}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- t1 u) t1)))
   (if (<= u -1.02e-55)
     (/ (/ v t_1) (+ t1 u))
     (if (<= u 2.1e-91)
       (/ (- v) t1)
       (if (<= u 4.7e-32)
         (/ v (* (+ t1 u) t_1))
         (if (<= u 8.5e+58)
           (/ v (- (* u -2.0) t1))
           (/ 1.0 (/ (- t1 u) (* t1 (/ v u))))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 - u) / t1;
	double tmp;
	if (u <= -1.02e-55) {
		tmp = (v / t_1) / (t1 + u);
	} else if (u <= 2.1e-91) {
		tmp = -v / t1;
	} else if (u <= 4.7e-32) {
		tmp = v / ((t1 + u) * t_1);
	} else if (u <= 8.5e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 1.0 / ((t1 - u) / (t1 * (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) :: t_1
    real(8) :: tmp
    t_1 = (t1 - u) / t1
    if (u <= (-1.02d-55)) then
        tmp = (v / t_1) / (t1 + u)
    else if (u <= 2.1d-91) then
        tmp = -v / t1
    else if (u <= 4.7d-32) then
        tmp = v / ((t1 + u) * t_1)
    else if (u <= 8.5d+58) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = 1.0d0 / ((t1 - u) / (t1 * (v / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 - u) / t1;
	double tmp;
	if (u <= -1.02e-55) {
		tmp = (v / t_1) / (t1 + u);
	} else if (u <= 2.1e-91) {
		tmp = -v / t1;
	} else if (u <= 4.7e-32) {
		tmp = v / ((t1 + u) * t_1);
	} else if (u <= 8.5e+58) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 - u) / t1
	tmp = 0
	if u <= -1.02e-55:
		tmp = (v / t_1) / (t1 + u)
	elif u <= 2.1e-91:
		tmp = -v / t1
	elif u <= 4.7e-32:
		tmp = v / ((t1 + u) * t_1)
	elif u <= 8.5e+58:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 - u) / t1)
	tmp = 0.0
	if (u <= -1.02e-55)
		tmp = Float64(Float64(v / t_1) / Float64(t1 + u));
	elseif (u <= 2.1e-91)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 4.7e-32)
		tmp = Float64(v / Float64(Float64(t1 + u) * t_1));
	elseif (u <= 8.5e+58)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(1.0 / Float64(Float64(t1 - u) / Float64(t1 * Float64(v / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 - u) / t1;
	tmp = 0.0;
	if (u <= -1.02e-55)
		tmp = (v / t_1) / (t1 + u);
	elseif (u <= 2.1e-91)
		tmp = -v / t1;
	elseif (u <= 4.7e-32)
		tmp = v / ((t1 + u) * t_1);
	elseif (u <= 8.5e+58)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = 1.0 / ((t1 - u) / (t1 * (v / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]}, If[LessEqual[u, -1.02e-55], N[(N[(v / t$95$1), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.1e-91], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 4.7e-32], N[(v / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.5e+58], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t1 - u), $MachinePrecision] / N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1.02e-55

   1. Initial program 79.0%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

      1. associate-*r/ 99.2%

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

      2. clear-num 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-un-lft-identity 99.2%

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

      5. frac-2neg 99.2%

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

      6. distribute-neg-in 99.2%

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

      7. add-sqr-sqrt 47.5%

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

      8. sqrt-unprod 83.5%

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

      9. sqr-neg 83.5%

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

      10. sqrt-unprod 44.1%

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

      11. add-sqr-sqrt 82.9%

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

      12. sub-neg 82.9%

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

      13. remove-double-neg 82.9%

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

   5. Applied egg-rr 82.9%

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

   if -1.02e-55 < u < 2.0999999999999999e-91

   1. Initial program 67.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if 2.0999999999999999e-91 < u < 4.70000000000000019e-32

   1. Initial program 89.4%

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

      1. clear-num 89.3%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. frac-2neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. add-sqr-sqrt 77.4%

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

      7. sqrt-unprod 88.7%

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

      8. sqr-neg 88.7%

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

      9. sqrt-unprod 11.3%

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

      10. add-sqr-sqrt 88.7%

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

      11. sub-neg 88.7%

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

      12. remove-double-neg 88.7%

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

   5. Applied egg-rr 88.7%

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

   if 4.70000000000000019e-32 < u < 8.50000000000000015e58

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   if 8.50000000000000015e58 < u

   1. Initial program 84.9%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. clear-num 98.0%

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

      3. frac-2neg 98.0%

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

      4. frac-times 94.2%

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

      5. *-un-lft-identity 94.2%

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

      6. remove-double-neg 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. add-sqr-sqrt 55.6%

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

      9. sqrt-unprod 90.4%

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

      10. sqr-neg 90.4%

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

      11. sqrt-unprod 34.8%

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

      12. add-sqr-sqrt 90.4%

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

      13. sub-neg 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in t1 around 0 90.4%

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

      1. clear-num 90.3%

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

      2. inv-pow 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-/l* 93.8%

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

      5. div-inv 93.8%

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

      6. clear-num 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. unpow-1 93.9%

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

   10. Simplified 93.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{t1 \cdot \frac{v}{u}}}\\ \end{array} \]

Alternative 9: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{if}\;u \leq -4.1 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.35 \cdot 10^{-31} \lor \neg \left(u \leq 8.5 \cdot 10^{+58}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (+ t1 u)) (/ (- t1) u))))
   (if (<= u -4.1e-57)
     t_1
     (if (<= u 7.6e-88)
       (/ (- v) t1)
       (if (or (<= u 2.35e-31) (not (<= u 8.5e+58)))
         t_1
         (/ v (- (* u -2.0) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) * (-t1 / u);
	double tmp;
	if (u <= -4.1e-57) {
		tmp = t_1;
	} else if (u <= 7.6e-88) {
		tmp = -v / t1;
	} else if ((u <= 2.35e-31) || !(u <= 8.5e+58)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / (t1 + u)) * (-t1 / u)
    if (u <= (-4.1d-57)) then
        tmp = t_1
    else if (u <= 7.6d-88) then
        tmp = -v / t1
    else if ((u <= 2.35d-31) .or. (.not. (u <= 8.5d+58))) then
        tmp = t_1
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / (t1 + u)) * (-t1 / u);
	double tmp;
	if (u <= -4.1e-57) {
		tmp = t_1;
	} else if (u <= 7.6e-88) {
		tmp = -v / t1;
	} else if ((u <= 2.35e-31) || !(u <= 8.5e+58)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (t1 + u)) * (-t1 / u)
	tmp = 0
	if u <= -4.1e-57:
		tmp = t_1
	elif u <= 7.6e-88:
		tmp = -v / t1
	elif (u <= 2.35e-31) or not (u <= 8.5e+58):
		tmp = t_1
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u))
	tmp = 0.0
	if (u <= -4.1e-57)
		tmp = t_1;
	elseif (u <= 7.6e-88)
		tmp = Float64(Float64(-v) / t1);
	elseif ((u <= 2.35e-31) || !(u <= 8.5e+58))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / (t1 + u)) * (-t1 / u);
	tmp = 0.0;
	if (u <= -4.1e-57)
		tmp = t_1;
	elseif (u <= 7.6e-88)
		tmp = -v / t1;
	elseif ((u <= 2.35e-31) || ~((u <= 8.5e+58)))
		tmp = t_1;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.1e-57], t$95$1, If[LessEqual[u, 7.6e-88], N[((-v) / t1), $MachinePrecision], If[Or[LessEqual[u, 2.35e-31], N[Not[LessEqual[u, 8.5e+58]], $MachinePrecision]], t$95$1, N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.1000000000000001e-57 or 7.60000000000000022e-88 < u < 2.34999999999999993e-31 or 8.50000000000000015e58 < u

   1. Initial program 81.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 86.1%

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

      1. associate-*r/ 86.1%

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

      2. mul-1-neg 86.1%

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

   6. Simplified 86.1%

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

   if -4.1000000000000001e-57 < u < 7.60000000000000022e-88

   1. Initial program 67.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   6. Simplified 86.5%

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

   if 2.34999999999999993e-31 < u < 8.50000000000000015e58

   1. Initial program 95.0%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 96.5%

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

      4. associate-/l/ 96.5%

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

      5. +-commutative 96.5%

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

      6. remove-double-neg 96.5%

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

      7. unsub-neg 96.5%

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

      8. div-sub 96.5%

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

      9. sub-neg 96.5%

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

      10. *-inverses 96.5%

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

      11. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. *-commutative 67.1%

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

   6. Simplified 67.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.35 \cdot 10^{-31} \lor \neg \left(u \leq 8.5 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 10: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -3.5 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (/ t1 u) (- t1 u)))) (t_2 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -3.5e-27)
     t_2
     (if (<= t1 4.5e-77)
       t_1
       (if (<= t1 7e-41)
         t_2
         (if (<= t1 7.6e-6) t_1 (/ (- v) (+ t1 (* u -2.0)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / (t1 - u));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -3.5e-27) {
		tmp = t_2;
	} else if (t1 <= 4.5e-77) {
		tmp = t_1;
	} else if (t1 <= 7e-41) {
		tmp = t_2;
	} else if (t1 <= 7.6e-6) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v * ((t1 / u) / (t1 - u))
    t_2 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-3.5d-27)) then
        tmp = t_2
    else if (t1 <= 4.5d-77) then
        tmp = t_1
    else if (t1 <= 7d-41) then
        tmp = t_2
    else if (t1 <= 7.6d-6) then
        tmp = t_1
    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 t_1 = v * ((t1 / u) / (t1 - u));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -3.5e-27) {
		tmp = t_2;
	} else if (t1 <= 4.5e-77) {
		tmp = t_1;
	} else if (t1 <= 7e-41) {
		tmp = t_2;
	} else if (t1 <= 7.6e-6) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + (u * -2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * ((t1 / u) / (t1 - u))
	t_2 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -3.5e-27:
		tmp = t_2
	elif t1 <= 4.5e-77:
		tmp = t_1
	elif t1 <= 7e-41:
		tmp = t_2
	elif t1 <= 7.6e-6:
		tmp = t_1
	else:
		tmp = -v / (t1 + (u * -2.0))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(t1 / u) / Float64(t1 - u)))
	t_2 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -3.5e-27)
		tmp = t_2;
	elseif (t1 <= 4.5e-77)
		tmp = t_1;
	elseif (t1 <= 7e-41)
		tmp = t_2;
	elseif (t1 <= 7.6e-6)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * ((t1 / u) / (t1 - u));
	t_2 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -3.5e-27)
		tmp = t_2;
	elseif (t1 <= 4.5e-77)
		tmp = t_1;
	elseif (t1 <= 7e-41)
		tmp = t_2;
	elseif (t1 <= 7.6e-6)
		tmp = t_1;
	else
		tmp = -v / (t1 + (u * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(N[(t1 / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.5e-27], t$95$2, If[LessEqual[t1, 4.5e-77], t$95$1, If[LessEqual[t1, 7e-41], t$95$2, If[LessEqual[t1, 7.6e-6], t$95$1, N[((-v) / N[(t1 + N[(u * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.5000000000000001e-27 or 4.5000000000000001e-77 < t1 < 6.9999999999999999e-41

   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. associate-/r* 88.0%

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

      2. *-commutative 88.0%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 97.2%

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

      5. +-commutative 97.2%

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

      6. remove-double-neg 97.2%

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

      7. unsub-neg 97.2%

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

      8. div-sub 97.2%

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

      9. sub-neg 97.2%

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

      10. *-inverses 97.2%

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

      11. metadata-eval 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. *-commutative 78.2%

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

   6. Simplified 78.2%

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

   if -3.5000000000000001e-27 < t1 < 4.5000000000000001e-77 or 6.9999999999999999e-41 < t1 < 7.6000000000000001e-6

   1. Initial program 82.5%

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

      1. times-frac 95.3%

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

   3. Simplified 95.3%

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

      1. *-commutative 95.3%

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

      2. clear-num 95.2%

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

      3. frac-2neg 95.2%

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

      4. frac-times 89.1%

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

      5. *-un-lft-identity 89.1%

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

      6. remove-double-neg 89.1%

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

      7. distribute-neg-in 89.1%

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

      8. add-sqr-sqrt 47.7%

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

      9. sqrt-unprod 79.3%

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

      10. sqr-neg 79.3%

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

      11. sqrt-unprod 33.2%

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

      12. add-sqr-sqrt 70.3%

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

      13. sub-neg 70.3%

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

   5. Applied egg-rr 70.3%

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

   6. Taylor expanded in t1 around 0 73.2%

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

   7. Taylor expanded in v around 0 73.0%

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

      1. associate-*l/ 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-/r* 77.2%

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

   9. Simplified 77.2%

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

   if 7.6000000000000001e-6 < t1

   1. Initial program 65.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. clear-num 99.8%

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

      2. frac-2neg 99.8%

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

      3. frac-times 96.9%

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

      4. *-un-lft-identity 96.9%

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

      5. frac-2neg 96.9%

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

      6. distribute-neg-in 96.9%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 37.5%

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

      9. sqr-neg 37.5%

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

      10. sqrt-unprod 38.6%

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

      11. add-sqr-sqrt 38.6%

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

      12. sub-neg 38.6%

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

      13. remove-double-neg 38.6%

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

      14. distribute-neg-in 38.6%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 71.8%

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

      17. sqr-neg 71.8%

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

      18. sqrt-unprod 95.9%

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

      19. add-sqr-sqrt 96.3%

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

      20. sub-neg 96.3%

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

   5. Applied egg-rr 96.3%

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

      1. associate-*l/ 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in t1 around inf 88.1%

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

      1. *-commutative 88.1%

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

   10. Simplified 88.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{-77}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot -2}\\ \end{array} \]

Alternative 11: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. times-frac 97.7%

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

3. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 12: 59.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.2e+194)
   (/ v u)
   (if (<= u 4.7e+126) (/ (- v) t1) (* v (/ t1 (* t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+194) {
		tmp = v / u;
	} else if (u <= 4.7e+126) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (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 <= (-1.2d+194)) then
        tmp = v / u
    else if (u <= 4.7d+126) then
        tmp = -v / t1
    else
        tmp = v * (t1 / (t1 * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+194) {
		tmp = v / u;
	} else if (u <= 4.7e+126) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (t1 * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.2e+194:
		tmp = v / u
	elif u <= 4.7e+126:
		tmp = -v / t1
	else:
		tmp = v * (t1 / (t1 * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.2e+194)
		tmp = Float64(v / u);
	elseif (u <= 4.7e+126)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v * Float64(t1 / Float64(t1 * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.2e+194)
		tmp = v / u;
	elseif (u <= 4.7e+126)
		tmp = -v / t1;
	else
		tmp = v * (t1 / (t1 * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.2e+194], N[(v / u), $MachinePrecision], If[LessEqual[u, 4.7e+126], N[((-v) / t1), $MachinePrecision], N[(v * N[(t1 / N[(t1 * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.2e194

   1. Initial program 87.8%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. frac-2neg 99.6%

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

      4. frac-times 92.8%

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

      5. *-un-lft-identity 92.8%

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

      6. remove-double-neg 92.8%

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

      7. distribute-neg-in 92.8%

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

      8. add-sqr-sqrt 42.3%

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

      9. sqrt-unprod 92.8%

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

      10. sqr-neg 92.8%

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

      11. sqrt-unprod 50.5%

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

      12. add-sqr-sqrt 92.8%

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

      13. sub-neg 92.8%

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

   5. Applied egg-rr 92.8%

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

   6. Taylor expanded in t1 around 0 92.8%

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

   7. Taylor expanded in t1 around inf 52.6%

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

   if -1.2e194 < u < 4.6999999999999999e126

   1. Initial program 74.6%

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

      1. times-frac 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t1 around inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. neg-mul-1 62.8%

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

   6. Simplified 62.8%

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

   if 4.6999999999999999e126 < u

   1. Initial program 86.9%

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

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. frac-times 96.9%

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

      5. *-un-lft-identity 96.9%

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

      6. remove-double-neg 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. add-sqr-sqrt 61.7%

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

      9. sqrt-unprod 96.9%

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

      10. sqr-neg 96.9%

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

      11. sqrt-unprod 35.2%

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

      12. add-sqr-sqrt 96.9%

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

      13. sub-neg 96.9%

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

   5. Applied egg-rr 96.9%

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

   6. Taylor expanded in t1 around 0 96.9%

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

   7. Taylor expanded in u around 0 48.8%

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

      1. associate-/r/ 46.8%

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

   9. Applied egg-rr 46.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \end{array} \]

Alternative 13: 59.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+127}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.2e+194)
   (/ v u)
   (if (<= u 6e+127) (/ (- v) t1) (/ t1 (* t1 (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+194) {
		tmp = v / u;
	} else if (u <= 6e+127) {
		tmp = -v / t1;
	} else {
		tmp = t1 / (t1 * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.2d+194)) then
        tmp = v / u
    else if (u <= 6d+127) then
        tmp = -v / t1
    else
        tmp = t1 / (t1 * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+194) {
		tmp = v / u;
	} else if (u <= 6e+127) {
		tmp = -v / t1;
	} else {
		tmp = t1 / (t1 * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.2e+194:
		tmp = v / u
	elif u <= 6e+127:
		tmp = -v / t1
	else:
		tmp = t1 / (t1 * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.2e+194)
		tmp = Float64(v / u);
	elseif (u <= 6e+127)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(t1 * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.2e+194)
		tmp = v / u;
	elseif (u <= 6e+127)
		tmp = -v / t1;
	else
		tmp = t1 / (t1 * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.2e+194], N[(v / u), $MachinePrecision], If[LessEqual[u, 6e+127], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(t1 * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.2e194

   1. Initial program 87.8%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. frac-2neg 99.6%

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

      4. frac-times 92.8%

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

      5. *-un-lft-identity 92.8%

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

      6. remove-double-neg 92.8%

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

      7. distribute-neg-in 92.8%

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

      8. add-sqr-sqrt 42.3%

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

      9. sqrt-unprod 92.8%

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

      10. sqr-neg 92.8%

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

      11. sqrt-unprod 50.5%

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

      12. add-sqr-sqrt 92.8%

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

      13. sub-neg 92.8%

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

   5. Applied egg-rr 92.8%

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

   6. Taylor expanded in t1 around 0 92.8%

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

   7. Taylor expanded in t1 around inf 52.6%

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

   if -1.2e194 < u < 6.0000000000000005e127

   1. Initial program 74.6%

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

      1. times-frac 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t1 around inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. neg-mul-1 62.8%

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

   6. Simplified 62.8%

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

   if 6.0000000000000005e127 < u

   1. Initial program 86.9%

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

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. frac-times 96.9%

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

      5. *-un-lft-identity 96.9%

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

      6. remove-double-neg 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. add-sqr-sqrt 61.7%

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

      9. sqrt-unprod 96.9%

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

      10. sqr-neg 96.9%

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

      11. sqrt-unprod 35.2%

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

      12. add-sqr-sqrt 96.9%

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

      13. sub-neg 96.9%

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

   5. Applied egg-rr 96.9%

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

   6. Taylor expanded in t1 around 0 96.9%

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

   7. Taylor expanded in u around 0 48.8%

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

      1. associate-*r/ 50.6%

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

   9. Simplified 50.6%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{+127}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 14: 58.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.3e+195)
   (/ v u)
   (if (<= u 2.9e+130) (/ (- v) t1) (/ (* v -0.5) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e+195) {
		tmp = v / u;
	} else if (u <= 2.9e+130) {
		tmp = -v / t1;
	} else {
		tmp = (v * -0.5) / 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 <= (-1.3d+195)) then
        tmp = v / u
    else if (u <= 2.9d+130) then
        tmp = -v / t1
    else
        tmp = (v * (-0.5d0)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e+195) {
		tmp = v / u;
	} else if (u <= 2.9e+130) {
		tmp = -v / t1;
	} else {
		tmp = (v * -0.5) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.3e+195:
		tmp = v / u
	elif u <= 2.9e+130:
		tmp = -v / t1
	else:
		tmp = (v * -0.5) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.3e+195)
		tmp = Float64(v / u);
	elseif (u <= 2.9e+130)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * -0.5) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.3e+195)
		tmp = v / u;
	elseif (u <= 2.9e+130)
		tmp = -v / t1;
	else
		tmp = (v * -0.5) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.3e+195], N[(v / u), $MachinePrecision], If[LessEqual[u, 2.9e+130], N[((-v) / t1), $MachinePrecision], N[(N[(v * -0.5), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.30000000000000001e195

   1. Initial program 87.8%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. frac-2neg 99.6%

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

      4. frac-times 92.8%

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

      5. *-un-lft-identity 92.8%

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

      6. remove-double-neg 92.8%

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

      7. distribute-neg-in 92.8%

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

      8. add-sqr-sqrt 42.3%

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

      9. sqrt-unprod 92.8%

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

      10. sqr-neg 92.8%

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

      11. sqrt-unprod 50.5%

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

      12. add-sqr-sqrt 92.8%

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

      13. sub-neg 92.8%

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

   5. Applied egg-rr 92.8%

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

   6. Taylor expanded in t1 around 0 92.8%

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

   7. Taylor expanded in t1 around inf 52.6%

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

   if -1.30000000000000001e195 < u < 2.8999999999999999e130

   1. Initial program 74.6%

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

      1. times-frac 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t1 around inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. neg-mul-1 62.8%

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

   6. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 2.8999999999999999e130 < u

   1. Initial program 86.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 94.7%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

      2. *-commutative 94.7%

         \[\leadsto \frac{\frac{\color{blue}{v \cdot \left(-t1\right)}}{t1 + u}}{t1 + u} \]

      3. associate-/l* 100.0%

         \[\leadsto \frac{\color{blue}{\frac{v}{\frac{t1 + u}{-t1}}}}{t1 + u} \]

      4. associate-/l/ 87.4%

         \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]

      5. +-commutative 87.4%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \frac{\color{blue}{u + t1}}{-t1}} \]

      6. remove-double-neg 87.4%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \frac{u + \color{blue}{\left(-\left(-t1\right)\right)}}{-t1}} \]

      7. unsub-neg 87.4%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \frac{\color{blue}{u - \left(-t1\right)}}{-t1}} \]

      8. div-sub 87.4%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{-t1} - \frac{-t1}{-t1}\right)}} \]

      9. sub-neg 87.4%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{-t1} + \left(-\frac{-t1}{-t1}\right)\right)}} \]

      10. *-inverses 87.4%

          \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\frac{u}{-t1} + \left(-\color{blue}{1}\right)\right)} \]

      11. metadata-eval 87.4%

          \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\frac{u}{-t1} + \color{blue}{-1}\right)} \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(\frac{u}{-t1} + -1\right)}} \]

   4. Taylor expanded in t1 around inf 44.8%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   5. Step-by-step derivation

      1. mul-1-neg 44.8%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 44.8%

         \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

      3. *-commutative 44.8%

         \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

   6. Simplified 44.8%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]

   7. Taylor expanded in u around inf 42.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   8. Step-by-step derivation

      1. associate-*r/ 42.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]

   9. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -0.5}{u}\\ \end{array} \]

Alternative 15: 58.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+195} \lor \neg \left(u \leq 3.5 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.4e+195) (not (<= u 3.5e+126))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.4e+195) || !(u <= 3.5e+126)) {
		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 <= (-5.4d+195)) .or. (.not. (u <= 3.5d+126))) 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 <= -5.4e+195) || !(u <= 3.5e+126)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.4e+195) or not (u <= 3.5e+126):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.4e+195) || !(u <= 3.5e+126))
		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 <= -5.4e+195) || ~((u <= 3.5e+126)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.4e+195], N[Not[LessEqual[u, 3.5e+126]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.4000000000000003e195 or 3.5000000000000003e126 < u

   1. Initial program 87.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      2. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \cdot \frac{-t1}{t1 + u} \]

      3. frac-2neg 99.8%

         \[\leadsto \frac{1}{\frac{t1 + u}{v}} \cdot \color{blue}{\frac{-\left(-t1\right)}{-\left(t1 + u\right)}} \]

      4. frac-times 95.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-\left(-t1\right)\right)}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)}} \]

      5. *-un-lft-identity 95.2%

         \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

      6. remove-double-neg 95.2%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

      7. distribute-neg-in 95.2%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(\left(-t1\right) + \left(-u\right)\right)}} \]

      8. add-sqr-sqrt 53.7%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)\right)} \]

      9. sqrt-unprod 95.2%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)\right)} \]

      10. sqr-neg 95.2%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)\right)} \]

      11. sqrt-unprod 41.5%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)\right)} \]

      12. add-sqr-sqrt 95.2%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{t1} + \left(-u\right)\right)} \]

      13. sub-neg 95.2%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(t1 - u\right)}} \]

   5. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}} \]

   6. Taylor expanded in t1 around 0 95.2%

      \[\leadsto \frac{t1}{\color{blue}{\frac{u}{v}} \cdot \left(t1 - u\right)} \]

   7. Taylor expanded in t1 around inf 46.5%

      \[\leadsto \color{blue}{\frac{v}{u}} \]

   if -5.4000000000000003e195 < u < 3.5000000000000003e126

   1. Initial program 74.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 97.0%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 62.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   5. Step-by-step derivation

      1. associate-*r/ 62.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 62.8%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   6. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+195} \lor \neg \left(u \leq 3.5 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 16: 23.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6.7 \cdot 10^{+106} \lor \neg \left(t1 \leq 2.8 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6.7e+106) (not (<= t1 2.8e+105))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6.7e+106) || !(t1 <= 2.8e+105)) {
		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 <= (-6.7d+106)) .or. (.not. (t1 <= 2.8d+105))) 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 <= -6.7e+106) || !(t1 <= 2.8e+105)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6.7e+106) or not (t1 <= 2.8e+105):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6.7e+106) || !(t1 <= 2.8e+105))
		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 <= -6.7e+106) || ~((t1 <= 2.8e+105)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6.7e+106], N[Not[LessEqual[t1, 2.8e+105]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.7e106 or 2.8000000000000001e105 < t1

   1. Initial program 59.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \cdot \frac{-t1}{t1 + u} \]

      3. frac-2neg 99.6%

         \[\leadsto \frac{1}{\frac{t1 + u}{v}} \cdot \color{blue}{\frac{-\left(-t1\right)}{-\left(t1 + u\right)}} \]

      4. frac-times 68.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-\left(-t1\right)\right)}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)}} \]

      5. *-un-lft-identity 68.0%

         \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

      6. remove-double-neg 68.0%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

      7. distribute-neg-in 68.0%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(\left(-t1\right) + \left(-u\right)\right)}} \]

      8. add-sqr-sqrt 32.9%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)\right)} \]

      9. sqrt-unprod 49.9%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)\right)} \]

      10. sqr-neg 49.9%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)\right)} \]

      11. sqrt-unprod 25.0%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)\right)} \]

      12. add-sqr-sqrt 48.3%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{t1} + \left(-u\right)\right)} \]

      13. sub-neg 48.3%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(t1 - u\right)}} \]

   5. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}} \]

   6. Taylor expanded in t1 around inf 41.7%

      \[\leadsto \color{blue}{\frac{v}{t1}} \]

   if -6.7e106 < t1 < 2.8000000000000001e105

   1. Initial program 84.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      2. clear-num 96.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \cdot \frac{-t1}{t1 + u} \]

      3. frac-2neg 96.7%

         \[\leadsto \frac{1}{\frac{t1 + u}{v}} \cdot \color{blue}{\frac{-\left(-t1\right)}{-\left(t1 + u\right)}} \]

      4. frac-times 89.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-\left(-t1\right)\right)}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)}} \]

      5. *-un-lft-identity 89.8%

         \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

      6. remove-double-neg 89.8%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

      7. distribute-neg-in 89.8%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(\left(-t1\right) + \left(-u\right)\right)}} \]

      8. add-sqr-sqrt 46.7%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)\right)} \]

      9. sqrt-unprod 71.4%

         \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)\right)} \]

      10. sqr-neg 71.4%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)\right)} \]

      11. sqrt-unprod 25.7%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)\right)} \]

      12. add-sqr-sqrt 57.6%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{t1} + \left(-u\right)\right)} \]

      13. sub-neg 57.6%

          \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(t1 - u\right)}} \]

   5. Applied egg-rr 57.6%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}} \]

   6. Taylor expanded in t1 around 0 59.8%

      \[\leadsto \frac{t1}{\color{blue}{\frac{u}{v}} \cdot \left(t1 - u\right)} \]

   7. Taylor expanded in t1 around inf 17.3%

      \[\leadsto \color{blue}{\frac{v}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.7 \cdot 10^{+106} \lor \neg \left(t1 \leq 2.8 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 17: 62.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (* u -2.0) t1)))

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / ((u * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(u * -2.0) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/r* 87.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   2. *-commutative 87.6%

      \[\leadsto \frac{\frac{\color{blue}{v \cdot \left(-t1\right)}}{t1 + u}}{t1 + u} \]

   3. associate-/l* 98.6%

      \[\leadsto \frac{\color{blue}{\frac{v}{\frac{t1 + u}{-t1}}}}{t1 + u} \]

   4. associate-/l/ 95.4%

      \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]

   5. +-commutative 95.4%

      \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \frac{\color{blue}{u + t1}}{-t1}} \]

   6. remove-double-neg 95.4%

      \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \frac{u + \color{blue}{\left(-\left(-t1\right)\right)}}{-t1}} \]

   7. unsub-neg 95.4%

      \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \frac{\color{blue}{u - \left(-t1\right)}}{-t1}} \]

   8. div-sub 95.4%

      \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{-t1} - \frac{-t1}{-t1}\right)}} \]

   9. sub-neg 95.4%

      \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{-t1} + \left(-\frac{-t1}{-t1}\right)\right)}} \]

   10. *-inverses 95.4%

       \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\frac{u}{-t1} + \left(-\color{blue}{1}\right)\right)} \]

   11. metadata-eval 95.4%

       \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\frac{u}{-t1} + \color{blue}{-1}\right)} \]

3. Simplified 95.4%

   \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(\frac{u}{-t1} + -1\right)}} \]

4. Taylor expanded in t1 around inf 60.4%

   \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
5. Step-by-step derivation

   1. mul-1-neg 60.4%

      \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

   2. unsub-neg 60.4%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

   3. *-commutative 60.4%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

6. Simplified 60.4%

   \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]

7. Final simplification 60.4%

   \[\leadsto \frac{v}{u \cdot -2 - t1} \]

Alternative 18: 13.8% 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 77.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 97.7%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
4. Step-by-step derivation

   1. *-commutative 97.7%

      \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

   2. clear-num 97.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \cdot \frac{-t1}{t1 + u} \]

   3. frac-2neg 97.5%

      \[\leadsto \frac{1}{\frac{t1 + u}{v}} \cdot \color{blue}{\frac{-\left(-t1\right)}{-\left(t1 + u\right)}} \]

   4. frac-times 83.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(-\left(-t1\right)\right)}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)}} \]

   5. *-un-lft-identity 83.8%

      \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

   6. remove-double-neg 83.8%

      \[\leadsto \frac{\color{blue}{t1}}{\frac{t1 + u}{v} \cdot \left(-\left(t1 + u\right)\right)} \]

   7. distribute-neg-in 83.8%

      \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(\left(-t1\right) + \left(-u\right)\right)}} \]

   8. add-sqr-sqrt 42.9%

      \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)\right)} \]

   9. sqrt-unprod 65.4%

      \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)\right)} \]

   10. sqr-neg 65.4%

       \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)\right)} \]

   11. sqrt-unprod 25.5%

       \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)\right)} \]

   12. add-sqr-sqrt 55.0%

       \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \left(\color{blue}{t1} + \left(-u\right)\right)} \]

   13. sub-neg 55.0%

       \[\leadsto \frac{t1}{\frac{t1 + u}{v} \cdot \color{blue}{\left(t1 - u\right)}} \]

5. Applied egg-rr 55.0%

   \[\leadsto \color{blue}{\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}} \]

6. Taylor expanded in t1 around inf 13.9%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]

7. Final simplification 13.9%

   \[\leadsto \frac{v}{t1} \]

Reproduce

herbie shell --seed 2023331 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))