Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 12.9s

Alternatives: 16

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: 98.1% accurate, 1.0× speedup

Math

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

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(t1 * Float64(Float64(-v) / 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 * N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.7%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 97.8%

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

3. Simplified 97.8%

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

   1. div-inv 97.5%

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

   2. clear-num 98.1%

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

   3. add-sqr-sqrt 50.3%

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

   4. sqrt-unprod 46.0%

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

   5. sqr-neg 46.0%

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

   6. sqrt-unprod 18.2%

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

   7. add-sqr-sqrt 35.1%

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

   8. distribute-lft-neg-in 35.1%

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

   9. distribute-rgt-neg-in 35.1%

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

   10. add-sqr-sqrt 18.2%

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

   11. sqrt-unprod 46.0%

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

   12. sqr-neg 46.0%

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

   13. sqrt-unprod 50.3%

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

   14. add-sqr-sqrt 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;u \leq -4.5 \cdot 10^{+177}:\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= u -4.5e+177)
     (/ t1 (* (+ t1 u) (/ (- u) v)))
     (if (<= u -4.8e-127)
       t_1
       (if (<= u 8.5e-162)
         (/ (- v) t1)
         (if (<= u 7.2e+103) t_1 (/ (/ t1 (+ t1 u)) (/ (- t1 u) v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (u <= -4.5e+177) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else if (u <= -4.8e-127) {
		tmp = t_1;
	} else if (u <= 8.5e-162) {
		tmp = -v / t1;
	} else if (u <= 7.2e+103) {
		tmp = t_1;
	} else {
		tmp = (t1 / (t1 + u)) / ((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) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (u <= (-4.5d+177)) then
        tmp = t1 / ((t1 + u) * (-u / v))
    else if (u <= (-4.8d-127)) then
        tmp = t_1
    else if (u <= 8.5d-162) then
        tmp = -v / t1
    else if (u <= 7.2d+103) then
        tmp = t_1
    else
        tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
    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 (u <= -4.5e+177) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else if (u <= -4.8e-127) {
		tmp = t_1;
	} else if (u <= 8.5e-162) {
		tmp = -v / t1;
	} else if (u <= 7.2e+103) {
		tmp = t_1;
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if u <= -4.5e+177:
		tmp = t1 / ((t1 + u) * (-u / v))
	elif u <= -4.8e-127:
		tmp = t_1
	elif u <= 8.5e-162:
		tmp = -v / t1
	elif u <= 7.2e+103:
		tmp = t_1
	else:
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
	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 (u <= -4.5e+177)
		tmp = Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) / v)));
	elseif (u <= -4.8e-127)
		tmp = t_1;
	elseif (u <= 8.5e-162)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 7.2e+103)
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(Float64(t1 - u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (u <= -4.5e+177)
		tmp = t1 / ((t1 + u) * (-u / v));
	elseif (u <= -4.8e-127)
		tmp = t_1;
	elseif (u <= 8.5e-162)
		tmp = -v / t1;
	elseif (u <= 7.2e+103)
		tmp = t_1;
	else
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	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[u, -4.5e+177], N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -4.8e-127], t$95$1, If[LessEqual[u, 8.5e-162], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 7.2e+103], t$95$1, N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -4.4999999999999997e177

   1. Initial program 67.9%

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

      1. associate-*l/ 68.3%

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

      2. *-commutative 68.3%

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

   3. Simplified 68.3%

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

      1. associate-/r* 77.5%

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

      2. associate-*r/ 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-/r/ 99.8%

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

      5. div-inv 99.6%

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

      6. frac-2neg 99.6%

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

      7. frac-times 96.9%

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

      8. remove-double-neg 96.9%

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

      9. *-commutative 96.9%

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

      10. *-un-lft-identity 96.9%

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

      11. distribute-neg-frac 96.9%

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

      12. distribute-neg-in 96.9%

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

      13. add-sqr-sqrt 41.9%

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

      14. sqrt-unprod 93.8%

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

      15. sqr-neg 93.8%

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

      16. sqrt-unprod 54.6%

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

      17. add-sqr-sqrt 96.6%

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

      18. sub-neg 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in t1 around 0 96.6%

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

      1. neg-mul-1 96.6%

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

      2. distribute-neg-frac 96.6%

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

   8. Simplified 96.6%

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

   if -4.4999999999999997e177 < u < -4.79999999999999964e-127 or 8.49999999999999955e-162 < u < 7.20000000000000033e103

   1. Initial program 77.1%

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

      1. associate-*l/ 87.1%

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

      2. *-commutative 87.1%

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

   3. Simplified 87.1%

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

   if -4.79999999999999964e-127 < u < 8.49999999999999955e-162

   1. Initial program 68.3%

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

      1. associate-*l/ 71.5%

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

      2. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in t1 around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   6. Simplified 91.3%

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

   if 7.20000000000000033e103 < u

   1. Initial program 78.4%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. associate-/r* 80.5%

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

      2. associate-*r/ 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-/r/ 98.6%

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

      5. div-inv 98.7%

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

      6. clear-num 98.6%

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

      7. associate-/r/ 98.6%

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

      8. clear-num 99.8%

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

      9. *-commutative 99.8%

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

      10. clear-num 98.6%

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

      11. div-inv 98.7%

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

      12. frac-2neg 98.7%

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

      13. remove-double-neg 98.7%

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

      14. associate-*l/ 97.6%

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

   5. Applied egg-rr 95.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+177}:\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \]

Alternative 3: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{t1}{-\frac{u}{\frac{v}{u}}}\\ \mathbf{if}\;t1 \leq -190000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;v \cdot \left(-\frac{\frac{t1}{u}}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (/ t1 (- (/ u (/ v u))))))
   (if (<= t1 -190000000.0)
     t_1
     (if (<= t1 -6.2e-47)
       t_2
       (if (<= t1 -8e-77)
         t_1
         (if (<= t1 -1.25e-152)
           t_2
           (if (<= t1 7.5e-17) (* v (- (/ (/ t1 u) u))) t_1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = t1 / -(u / (v / u));
	double tmp;
	if (t1 <= -190000000.0) {
		tmp = t_1;
	} else if (t1 <= -6.2e-47) {
		tmp = t_2;
	} else if (t1 <= -8e-77) {
		tmp = t_1;
	} else if (t1 <= -1.25e-152) {
		tmp = t_2;
	} else if (t1 <= 7.5e-17) {
		tmp = v * -((t1 / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = t1 / -(u / (v / u))
    if (t1 <= (-190000000.0d0)) then
        tmp = t_1
    else if (t1 <= (-6.2d-47)) then
        tmp = t_2
    else if (t1 <= (-8d-77)) then
        tmp = t_1
    else if (t1 <= (-1.25d-152)) then
        tmp = t_2
    else if (t1 <= 7.5d-17) then
        tmp = v * -((t1 / u) / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = t1 / -(u / (v / u));
	double tmp;
	if (t1 <= -190000000.0) {
		tmp = t_1;
	} else if (t1 <= -6.2e-47) {
		tmp = t_2;
	} else if (t1 <= -8e-77) {
		tmp = t_1;
	} else if (t1 <= -1.25e-152) {
		tmp = t_2;
	} else if (t1 <= 7.5e-17) {
		tmp = v * -((t1 / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = t1 / -(u / (v / u))
	tmp = 0
	if t1 <= -190000000.0:
		tmp = t_1
	elif t1 <= -6.2e-47:
		tmp = t_2
	elif t1 <= -8e-77:
		tmp = t_1
	elif t1 <= -1.25e-152:
		tmp = t_2
	elif t1 <= 7.5e-17:
		tmp = v * -((t1 / u) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(t1 / Float64(-Float64(u / Float64(v / u))))
	tmp = 0.0
	if (t1 <= -190000000.0)
		tmp = t_1;
	elseif (t1 <= -6.2e-47)
		tmp = t_2;
	elseif (t1 <= -8e-77)
		tmp = t_1;
	elseif (t1 <= -1.25e-152)
		tmp = t_2;
	elseif (t1 <= 7.5e-17)
		tmp = Float64(v * Float64(-Float64(Float64(t1 / u) / u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = t1 / -(u / (v / u));
	tmp = 0.0;
	if (t1 <= -190000000.0)
		tmp = t_1;
	elseif (t1 <= -6.2e-47)
		tmp = t_2;
	elseif (t1 <= -8e-77)
		tmp = t_1;
	elseif (t1 <= -1.25e-152)
		tmp = t_2;
	elseif (t1 <= 7.5e-17)
		tmp = v * -((t1 / u) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 / (-N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t1, -190000000.0], t$95$1, If[LessEqual[t1, -6.2e-47], t$95$2, If[LessEqual[t1, -8e-77], t$95$1, If[LessEqual[t1, -1.25e-152], t$95$2, If[LessEqual[t1, 7.5e-17], N[(v * (-N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision])), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.9e8 or -6.1999999999999996e-47 < t1 < -7.9999999999999994e-77 or 7.49999999999999984e-17 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -1.9e8 < t1 < -6.1999999999999996e-47 or -7.9999999999999994e-77 < t1 < -1.2499999999999999e-152

   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. associate-*l/ 80.9%

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

      2. *-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. associate-/r* 80.8%

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

      2. associate-*r/ 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-/r/ 99.4%

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

      5. div-inv 99.3%

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

      6. frac-2neg 99.3%

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

      7. frac-times 99.5%

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

      8. remove-double-neg 99.5%

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

      9. *-commutative 99.5%

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

      10. *-un-lft-identity 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. add-sqr-sqrt 99.4%

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

      14. sqrt-unprod 99.5%

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

      15. sqr-neg 99.5%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 83.2%

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

      18. sub-neg 83.2%

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

   5. Applied egg-rr 83.2%

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

   6. Taylor expanded in t1 around 0 71.0%

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

      1. mul-1-neg 71.0%

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

      2. unpow2 71.0%

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

      3. associate-/l* 84.6%

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

      4. distribute-neg-frac 84.6%

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

   8. Simplified 84.6%

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

   if -1.2499999999999999e-152 < t1 < 7.49999999999999984e-17

   1. Initial program 81.1%

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

      1. associate-/r* 87.5%

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

      2. associate-/l* 95.4%

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

   3. Simplified 95.4%

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

      1. div-inv 94.7%

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

      2. clear-num 94.8%

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

      3. add-sqr-sqrt 60.6%

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

      4. sqrt-unprod 62.2%

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

      5. sqr-neg 62.2%

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

      6. sqrt-unprod 14.8%

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

      7. add-sqr-sqrt 38.7%

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

      8. distribute-lft-neg-in 38.7%

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

      9. distribute-rgt-neg-in 38.7%

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

      10. add-sqr-sqrt 14.8%

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

      11. sqrt-unprod 62.2%

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

      12. sqr-neg 62.2%

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

      13. sqrt-unprod 60.6%

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

      14. add-sqr-sqrt 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in t1 around 0 74.9%

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

      1. mul-1-neg 74.9%

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

      2. associate-*l/ 79.0%

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

      3. unpow2 79.0%

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

      4. *-commutative 79.0%

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

      5. distribute-lft-neg-in 79.0%

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

      6. associate-/r* 84.8%

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

   8. Simplified 84.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -190000000:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{t1}{-\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-152}:\\ \;\;\;\;\frac{t1}{-\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;v \cdot \left(-\frac{\frac{t1}{u}}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 4: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{-t1 \cdot \frac{v}{u}}{u}\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -6 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -7.4 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1} \cdot \frac{t1}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (/ (- (* t1 (/ v u))) u)))
   (if (<= t1 -4e+33)
     t_1
     (if (<= t1 -6e-47)
       t_2
       (if (<= t1 -7.4e-77)
         t_1
         (if (<= t1 3.2e-17) t_2 (* (/ (- v) t1) (/ t1 (+ t1 u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = -(t1 * (v / u)) / u;
	double tmp;
	if (t1 <= -4e+33) {
		tmp = t_1;
	} else if (t1 <= -6e-47) {
		tmp = t_2;
	} else if (t1 <= -7.4e-77) {
		tmp = t_1;
	} else if (t1 <= 3.2e-17) {
		tmp = t_2;
	} else {
		tmp = (-v / t1) * (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = -(t1 * (v / u)) / u
    if (t1 <= (-4d+33)) then
        tmp = t_1
    else if (t1 <= (-6d-47)) then
        tmp = t_2
    else if (t1 <= (-7.4d-77)) then
        tmp = t_1
    else if (t1 <= 3.2d-17) then
        tmp = t_2
    else
        tmp = (-v / t1) * (t1 / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = -(t1 * (v / u)) / u;
	double tmp;
	if (t1 <= -4e+33) {
		tmp = t_1;
	} else if (t1 <= -6e-47) {
		tmp = t_2;
	} else if (t1 <= -7.4e-77) {
		tmp = t_1;
	} else if (t1 <= 3.2e-17) {
		tmp = t_2;
	} else {
		tmp = (-v / t1) * (t1 / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = -(t1 * (v / u)) / u
	tmp = 0
	if t1 <= -4e+33:
		tmp = t_1
	elif t1 <= -6e-47:
		tmp = t_2
	elif t1 <= -7.4e-77:
		tmp = t_1
	elif t1 <= 3.2e-17:
		tmp = t_2
	else:
		tmp = (-v / t1) * (t1 / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(Float64(-Float64(t1 * Float64(v / u))) / u)
	tmp = 0.0
	if (t1 <= -4e+33)
		tmp = t_1;
	elseif (t1 <= -6e-47)
		tmp = t_2;
	elseif (t1 <= -7.4e-77)
		tmp = t_1;
	elseif (t1 <= 3.2e-17)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(-v) / t1) * Float64(t1 / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = -(t1 * (v / u)) / u;
	tmp = 0.0;
	if (t1 <= -4e+33)
		tmp = t_1;
	elseif (t1 <= -6e-47)
		tmp = t_2;
	elseif (t1 <= -7.4e-77)
		tmp = t_1;
	elseif (t1 <= 3.2e-17)
		tmp = t_2;
	else
		tmp = (-v / t1) * (t1 / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]) / u), $MachinePrecision]}, If[LessEqual[t1, -4e+33], t$95$1, If[LessEqual[t1, -6e-47], t$95$2, If[LessEqual[t1, -7.4e-77], t$95$1, If[LessEqual[t1, 3.2e-17], t$95$2, N[(N[((-v) / t1), $MachinePrecision] * N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.9999999999999998e33 or -6.00000000000000033e-47 < t1 < -7.39999999999999992e-77

   1. Initial program 70.5%

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

      1. associate-/r* 79.9%

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

      2. associate-/l* 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in t1 around inf 90.6%

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

      1. neg-mul-1 90.6%

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

   6. Simplified 90.6%

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

   if -3.9999999999999998e33 < t1 < -6.00000000000000033e-47 or -7.39999999999999992e-77 < t1 < 3.2000000000000002e-17

   1. Initial program 79.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around 0 93.8%

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

      1. +-commutative 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in t1 around 0 71.0%

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

      1. unpow2 71.0%

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

      2. associate-/r* 78.6%

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

      3. associate-*l/ 83.4%

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

      4. associate-*r/ 83.4%

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

      5. neg-mul-1 83.4%

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

      6. distribute-rgt-neg-in 83.4%

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

      7. associate-*l/ 78.6%

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

      8. associate-*r/ 82.1%

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

   9. Simplified 82.1%

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

   if 3.2000000000000002e-17 < t1

   1. Initial program 67.0%

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

      1. associate-*l/ 72.8%

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

      2. *-commutative 72.8%

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

   3. Simplified 72.8%

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

      1. associate-/r* 94.8%

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

      2. associate-*r/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r/ 99.6%

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

      5. div-inv 99.5%

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

      6. frac-2neg 99.5%

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

      7. frac-times 81.9%

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

      8. remove-double-neg 81.9%

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

      9. *-commutative 81.9%

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

      10. *-un-lft-identity 81.9%

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

      11. distribute-neg-frac 81.9%

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

      12. distribute-neg-in 81.9%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 43.8%

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

      15. sqr-neg 43.8%

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

      16. sqrt-unprod 44.9%

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

      17. add-sqr-sqrt 44.9%

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

      18. sub-neg 44.9%

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

   5. Applied egg-rr 44.9%

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

   6. Taylor expanded in t1 around inf 29.6%

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

      1. frac-2neg 29.6%

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

      2. neg-sub0 29.6%

         \[\leadsto \frac{\color{blue}{0 - t1}}{-\frac{t1}{v} \cdot \left(t1 + u\right)} \]

      3. div-sub 29.6%

         \[\leadsto \color{blue}{\frac{0}{-\frac{t1}{v} \cdot \left(t1 + u\right)} - \frac{t1}{-\frac{t1}{v} \cdot \left(t1 + u\right)}} \]

   8. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\frac{0}{t1 \cdot \frac{u - t1}{v}} - \frac{v}{t1} \cdot \frac{t1}{t1 + u}} \]
   9. Step-by-step derivation

      1. div0 79.8%

         \[\leadsto \color{blue}{0} - \frac{v}{t1} \cdot \frac{t1}{t1 + u} \]

      2. neg-sub0 79.8%

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

      3. distribute-lft-neg-in 79.8%

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

      4. distribute-frac-neg 79.8%

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

   10. Simplified 79.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -6 \cdot 10^{-47}:\\ \;\;\;\;\frac{-t1 \cdot \frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq -7.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-t1 \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1} \cdot \frac{t1}{t1 + u}\\ \end{array} \]

Alternative 5: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -16000000 \lor \neg \left(t1 \leq -7.8 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 2.1 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -16000000.0)
         (not
          (or (<= t1 -7.8e-47) (and (not (<= t1 -8e-77)) (<= t1 2.1e-16)))))
   (/ (- v) (+ t1 u))
   (* t1 (/ v (* u (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -16000000.0) || !((t1 <= -7.8e-47) || (!(t1 <= -8e-77) && (t1 <= 2.1e-16)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * (v / (u * -u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-16000000.0d0)) .or. (.not. (t1 <= (-7.8d-47)) .or. (.not. (t1 <= (-8d-77))) .and. (t1 <= 2.1d-16))) then
        tmp = -v / (t1 + u)
    else
        tmp = t1 * (v / (u * -u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -16000000.0) || !((t1 <= -7.8e-47) || (!(t1 <= -8e-77) && (t1 <= 2.1e-16)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * (v / (u * -u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -16000000.0) or not ((t1 <= -7.8e-47) or (not (t1 <= -8e-77) and (t1 <= 2.1e-16))):
		tmp = -v / (t1 + u)
	else:
		tmp = t1 * (v / (u * -u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -16000000.0) || !((t1 <= -7.8e-47) || (!(t1 <= -8e-77) && (t1 <= 2.1e-16))))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -16000000.0) || ~(((t1 <= -7.8e-47) || (~((t1 <= -8e-77)) && (t1 <= 2.1e-16)))))
		tmp = -v / (t1 + u);
	else
		tmp = t1 * (v / (u * -u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -16000000.0], N[Not[Or[LessEqual[t1, -7.8e-47], And[N[Not[LessEqual[t1, -8e-77]], $MachinePrecision], LessEqual[t1, 2.1e-16]]]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6e7 or -7.79999999999999956e-47 < t1 < -7.9999999999999994e-77 or 2.1000000000000001e-16 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -1.6e7 < t1 < -7.79999999999999956e-47 or -7.9999999999999994e-77 < t1 < 2.1000000000000001e-16

   1. Initial program 79.7%

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

   2. Taylor expanded in t1 around 0 72.4%

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

      1. unpow2 72.4%

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

   4. Simplified 72.4%

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

      1. frac-2neg 72.4%

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

      2. div-inv 72.4%

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

      3. distribute-lft-neg-out 72.4%

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

      4. remove-double-neg 72.4%

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

      5. distribute-rgt-neg-in 72.4%

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

   6. Applied egg-rr 72.4%

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

      1. associate-*l* 72.3%

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

      2. associate-*r/ 72.3%

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

      3. *-rgt-identity 72.3%

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

   8. Simplified 72.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -16000000 \lor \neg \left(t1 \leq -7.8 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 2.1 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \end{array} \]

Alternative 6: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.6 \cdot 10^{+36} \lor \neg \left(t1 \leq -7 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 1.3 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1 \cdot \frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.6e+36)
         (not (or (<= t1 -7e-47) (and (not (<= t1 -8e-77)) (<= t1 1.3e-16)))))
   (/ (- v) (+ t1 u))
   (/ (- (* t1 (/ v u))) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.6e+36) || !((t1 <= -7e-47) || (!(t1 <= -8e-77) && (t1 <= 1.3e-16)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -(t1 * (v / u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-7.6d+36)) .or. (.not. (t1 <= (-7d-47)) .or. (.not. (t1 <= (-8d-77))) .and. (t1 <= 1.3d-16))) then
        tmp = -v / (t1 + u)
    else
        tmp = -(t1 * (v / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.6e+36) || !((t1 <= -7e-47) || (!(t1 <= -8e-77) && (t1 <= 1.3e-16)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -(t1 * (v / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.6e+36) or not ((t1 <= -7e-47) or (not (t1 <= -8e-77) and (t1 <= 1.3e-16))):
		tmp = -v / (t1 + u)
	else:
		tmp = -(t1 * (v / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.6e+36) || !((t1 <= -7e-47) || (!(t1 <= -8e-77) && (t1 <= 1.3e-16))))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-Float64(t1 * Float64(v / u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7.6e+36) || ~(((t1 <= -7e-47) || (~((t1 <= -8e-77)) && (t1 <= 1.3e-16)))))
		tmp = -v / (t1 + u);
	else
		tmp = -(t1 * (v / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.6e+36], N[Not[Or[LessEqual[t1, -7e-47], And[N[Not[LessEqual[t1, -8e-77]], $MachinePrecision], LessEqual[t1, 1.3e-16]]]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]) / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.6000000000000005e36 or -6.9999999999999996e-47 < t1 < -7.9999999999999994e-77 or 1.2999999999999999e-16 < t1

   1. Initial program 68.6%

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

      1. associate-/r* 78.7%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 84.8%

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

      1. neg-mul-1 84.8%

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

   6. Simplified 84.8%

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

   if -7.6000000000000005e36 < t1 < -6.9999999999999996e-47 or -7.9999999999999994e-77 < t1 < 1.2999999999999999e-16

   1. Initial program 79.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around 0 93.8%

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

      1. +-commutative 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in t1 around 0 71.0%

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

      1. unpow2 71.0%

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

      2. associate-/r* 78.6%

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

      3. associate-*l/ 83.4%

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

      4. associate-*r/ 83.4%

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

      5. neg-mul-1 83.4%

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

      6. distribute-rgt-neg-in 83.4%

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

      7. associate-*l/ 78.6%

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

      8. associate-*r/ 82.1%

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

   9. Simplified 82.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.6 \cdot 10^{+36} \lor \neg \left(t1 \leq -7 \cdot 10^{-47} \lor \neg \left(t1 \leq -8 \cdot 10^{-77}\right) \land t1 \leq 1.3 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1 \cdot \frac{v}{u}}{u}\\ \end{array} \]

Alternative 7: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -25500000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -6.1 \cdot 10^{-47}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77} \lor \neg \left(t1 \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -25500000.0)
     t_1
     (if (<= t1 -6.1e-47)
       (* t1 (/ v (* u (- u))))
       (if (or (<= t1 -8e-77) (not (<= t1 9.5e-17)))
         t_1
         (* (- v) (/ t1 (* u u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -25500000.0) {
		tmp = t_1;
	} else if (t1 <= -6.1e-47) {
		tmp = t1 * (v / (u * -u));
	} else if ((t1 <= -8e-77) || !(t1 <= 9.5e-17)) {
		tmp = t_1;
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-25500000.0d0)) then
        tmp = t_1
    else if (t1 <= (-6.1d-47)) then
        tmp = t1 * (v / (u * -u))
    else if ((t1 <= (-8d-77)) .or. (.not. (t1 <= 9.5d-17))) then
        tmp = t_1
    else
        tmp = -v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -25500000.0) {
		tmp = t_1;
	} else if (t1 <= -6.1e-47) {
		tmp = t1 * (v / (u * -u));
	} else if ((t1 <= -8e-77) || !(t1 <= 9.5e-17)) {
		tmp = t_1;
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -25500000.0:
		tmp = t_1
	elif t1 <= -6.1e-47:
		tmp = t1 * (v / (u * -u))
	elif (t1 <= -8e-77) or not (t1 <= 9.5e-17):
		tmp = t_1
	else:
		tmp = -v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -25500000.0)
		tmp = t_1;
	elseif (t1 <= -6.1e-47)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	elseif ((t1 <= -8e-77) || !(t1 <= 9.5e-17))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -25500000.0)
		tmp = t_1;
	elseif (t1 <= -6.1e-47)
		tmp = t1 * (v / (u * -u));
	elseif ((t1 <= -8e-77) || ~((t1 <= 9.5e-17)))
		tmp = t_1;
	else
		tmp = -v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -25500000.0], t$95$1, If[LessEqual[t1, -6.1e-47], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -8e-77], N[Not[LessEqual[t1, 9.5e-17]], $MachinePrecision]], t$95$1, N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.55e7 or -6.1e-47 < t1 < -7.9999999999999994e-77 or 9.50000000000000029e-17 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -2.55e7 < t1 < -6.1e-47

   1. Initial program 85.8%

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

   2. Taylor expanded in t1 around 0 76.1%

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

      1. unpow2 76.1%

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

   4. Simplified 76.1%

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

      1. frac-2neg 76.1%

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

      2. div-inv 76.0%

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

      3. distribute-lft-neg-out 76.0%

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

      4. remove-double-neg 76.0%

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

      5. distribute-rgt-neg-in 76.0%

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

   6. Applied egg-rr 76.0%

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

      1. associate-*l* 75.7%

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

      2. associate-*r/ 76.0%

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

      3. *-rgt-identity 76.0%

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

   8. Simplified 76.0%

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

   if -7.9999999999999994e-77 < t1 < 9.50000000000000029e-17

   1. Initial program 79.2%

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

      1. associate-*l/ 85.4%

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

      2. *-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in t1 around 0 77.5%

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

      1. associate-*r/ 77.5%

         \[\leadsto v \cdot \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \]

      2. neg-mul-1 77.5%

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

      3. unpow2 77.5%

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

   6. Simplified 77.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -25500000:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -6.1 \cdot 10^{-47}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77} \lor \neg \left(t1 \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 8: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -48000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77} \lor \neg \left(t1 \leq 4.9 \cdot 10^{-17}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -48000000.0)
     t_1
     (if (<= t1 -1.15e-46)
       (* t1 (/ v (* u (- u))))
       (if (or (<= t1 -8e-77) (not (<= t1 4.9e-17)))
         t_1
         (* v (/ (/ (- t1) u) u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -48000000.0) {
		tmp = t_1;
	} else if (t1 <= -1.15e-46) {
		tmp = t1 * (v / (u * -u));
	} else if ((t1 <= -8e-77) || !(t1 <= 4.9e-17)) {
		tmp = t_1;
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-48000000.0d0)) then
        tmp = t_1
    else if (t1 <= (-1.15d-46)) then
        tmp = t1 * (v / (u * -u))
    else if ((t1 <= (-8d-77)) .or. (.not. (t1 <= 4.9d-17))) then
        tmp = t_1
    else
        tmp = v * ((-t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -48000000.0) {
		tmp = t_1;
	} else if (t1 <= -1.15e-46) {
		tmp = t1 * (v / (u * -u));
	} else if ((t1 <= -8e-77) || !(t1 <= 4.9e-17)) {
		tmp = t_1;
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -48000000.0:
		tmp = t_1
	elif t1 <= -1.15e-46:
		tmp = t1 * (v / (u * -u))
	elif (t1 <= -8e-77) or not (t1 <= 4.9e-17):
		tmp = t_1
	else:
		tmp = v * ((-t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -48000000.0)
		tmp = t_1;
	elseif (t1 <= -1.15e-46)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	elseif ((t1 <= -8e-77) || !(t1 <= 4.9e-17))
		tmp = t_1;
	else
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -48000000.0)
		tmp = t_1;
	elseif (t1 <= -1.15e-46)
		tmp = t1 * (v / (u * -u));
	elseif ((t1 <= -8e-77) || ~((t1 <= 4.9e-17)))
		tmp = t_1;
	else
		tmp = v * ((-t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -48000000.0], t$95$1, If[LessEqual[t1, -1.15e-46], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -8e-77], N[Not[LessEqual[t1, 4.9e-17]], $MachinePrecision]], t$95$1, N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.8e7 or -1.15e-46 < t1 < -7.9999999999999994e-77 or 4.90000000000000012e-17 < t1

   1. Initial program 68.8%

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

      1. associate-/r* 79.3%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   6. Simplified 83.8%

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

   if -4.8e7 < t1 < -1.15e-46

   1. Initial program 85.8%

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

   2. Taylor expanded in t1 around 0 76.1%

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

      1. unpow2 76.1%

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

   4. Simplified 76.1%

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

      1. frac-2neg 76.1%

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

      2. div-inv 76.0%

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

      3. distribute-lft-neg-out 76.0%

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

      4. remove-double-neg 76.0%

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

      5. distribute-rgt-neg-in 76.0%

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

   6. Applied egg-rr 76.0%

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

      1. associate-*l* 75.7%

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

      2. associate-*r/ 76.0%

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

      3. *-rgt-identity 76.0%

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

   8. Simplified 76.0%

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

   if -7.9999999999999994e-77 < t1 < 4.90000000000000012e-17

   1. Initial program 79.2%

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

      1. associate-/r* 86.4%

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

      2. associate-/l* 95.9%

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

   3. Simplified 95.9%

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

      1. div-inv 95.3%

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

      2. clear-num 95.5%

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

      3. add-sqr-sqrt 52.5%

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

      4. sqrt-unprod 58.0%

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

      5. sqr-neg 58.0%

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

      6. sqrt-unprod 16.9%

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

      7. add-sqr-sqrt 37.7%

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

      8. distribute-lft-neg-in 37.7%

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

      9. distribute-rgt-neg-in 37.7%

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

      10. add-sqr-sqrt 16.9%

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

      11. sqrt-unprod 58.0%

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

      12. sqr-neg 58.0%

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

      13. sqrt-unprod 52.5%

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

      14. add-sqr-sqrt 95.5%

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

   5. Applied egg-rr 95.5%

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

   6. Taylor expanded in t1 around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-*l/ 77.5%

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

      3. unpow2 77.5%

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

      4. *-commutative 77.5%

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

      5. distribute-lft-neg-in 77.5%

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

      6. associate-/r* 82.5%

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

   8. Simplified 82.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -48000000:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-77} \lor \neg \left(t1 \leq 4.9 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \]

Alternative 9: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq 42:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.9e-115)
   (/ t1 (* (+ t1 u) (/ (- u) v)))
   (if (<= u 42.0) (/ (- v) t1) (/ (/ t1 (+ t1 u)) (/ (- t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-115) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else if (u <= 42.0) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / (t1 + u)) / ((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 <= (-2.9d-115)) then
        tmp = t1 / ((t1 + u) * (-u / v))
    else if (u <= 42.0d0) then
        tmp = -v / t1
    else
        tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-115) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else if (u <= 42.0) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.9e-115:
		tmp = t1 / ((t1 + u) * (-u / v))
	elif u <= 42.0:
		tmp = -v / t1
	else:
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.9e-115)
		tmp = Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) / v)));
	elseif (u <= 42.0)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(Float64(t1 - u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.9e-115)
		tmp = t1 / ((t1 + u) * (-u / v));
	elseif (u <= 42.0)
		tmp = -v / t1;
	else
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.9e-115], N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 42.0], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.8999999999999998e-115

   1. Initial program 70.4%

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

      1. associate-*l/ 77.7%

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

      2. *-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. associate-/r* 91.8%

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

      2. associate-*r/ 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-/r/ 95.9%

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

      5. div-inv 95.7%

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

      6. frac-2neg 95.7%

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

      7. frac-times 91.3%

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

      8. remove-double-neg 91.3%

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

      9. *-commutative 91.3%

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

      10. *-un-lft-identity 91.3%

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

      11. distribute-neg-frac 91.3%

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

      12. distribute-neg-in 91.3%

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

      13. add-sqr-sqrt 43.2%

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

      14. sqrt-unprod 83.6%

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

      15. sqr-neg 83.6%

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

      16. sqrt-unprod 44.6%

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

      17. add-sqr-sqrt 77.4%

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

      18. sub-neg 77.4%

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

   5. Applied egg-rr 77.4%

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

   6. Taylor expanded in t1 around 0 77.9%

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

      1. neg-mul-1 77.9%

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

      2. distribute-neg-frac 77.9%

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

   8. Simplified 77.9%

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

   if -2.8999999999999998e-115 < u < 42

   1. Initial program 72.0%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in t1 around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

   if 42 < u

   1. Initial program 81.9%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

   3. Simplified 80.4%

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

      1. associate-/r* 86.7%

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

      2. associate-*r/ 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/r/ 99.0%

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

      5. div-inv 99.0%

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

      6. clear-num 98.9%

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

      7. associate-/r/ 98.9%

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

      8. clear-num 99.8%

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

      9. *-commutative 99.8%

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

      10. clear-num 98.9%

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

      11. div-inv 99.0%

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

      12. frac-2neg 99.0%

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

      13. remove-double-neg 99.0%

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

      14. associate-*l/ 98.3%

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

   5. Applied egg-rr 90.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq 42:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \]

Alternative 10: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115} \lor \neg \left(u \leq 58\right):\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.9e-115) (not (<= u 58.0)))
   (/ t1 (* (+ t1 u) (/ (- u) v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e-115) || !(u <= 58.0)) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.9d-115)) .or. (.not. (u <= 58.0d0))) then
        tmp = t1 / ((t1 + u) * (-u / v))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e-115) || !(u <= 58.0)) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.9e-115) or not (u <= 58.0):
		tmp = t1 / ((t1 + u) * (-u / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.9e-115) || !(u <= 58.0))
		tmp = Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) / v)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.9e-115) || ~((u <= 58.0)))
		tmp = t1 / ((t1 + u) * (-u / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.9e-115], N[Not[LessEqual[u, 58.0]], $MachinePrecision]], N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.8999999999999998e-115 or 58 < u

   1. Initial program 75.0%

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

      1. associate-*l/ 78.8%

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

      2. *-commutative 78.8%

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

   3. Simplified 78.8%

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

      1. associate-/r* 89.8%

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

      2. associate-*r/ 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-/r/ 97.2%

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

      5. div-inv 97.0%

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

      6. frac-2neg 97.0%

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

      7. frac-times 90.6%

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

      8. remove-double-neg 90.6%

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

      9. *-commutative 90.6%

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

      10. *-un-lft-identity 90.6%

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

      11. distribute-neg-frac 90.6%

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

      12. distribute-neg-in 90.6%

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

      13. add-sqr-sqrt 46.7%

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

      14. sqrt-unprod 84.7%

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

      15. sqr-neg 84.7%

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

      16. sqrt-unprod 41.1%

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

      17. add-sqr-sqrt 80.3%

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

      18. sub-neg 80.3%

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

   5. Applied egg-rr 80.3%

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

   6. Taylor expanded in t1 around 0 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-neg-frac 80.6%

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

   8. Simplified 80.6%

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

   if -2.8999999999999998e-115 < u < 58

   1. Initial program 72.0%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in t1 around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115} \lor \neg \left(u \leq 58\right):\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq 45:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.9e-115)
   (/ t1 (* (+ t1 u) (/ (- u) v)))
   (if (<= u 45.0) (/ (- v) t1) (/ (/ (- t1) (/ u v)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-115) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else if (u <= 45.0) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (u / v)) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.9d-115)) then
        tmp = t1 / ((t1 + u) * (-u / v))
    else if (u <= 45.0d0) then
        tmp = -v / t1
    else
        tmp = (-t1 / (u / v)) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.9e-115) {
		tmp = t1 / ((t1 + u) * (-u / v));
	} else if (u <= 45.0) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (u / v)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.9e-115:
		tmp = t1 / ((t1 + u) * (-u / v))
	elif u <= 45.0:
		tmp = -v / t1
	else:
		tmp = (-t1 / (u / v)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.9e-115)
		tmp = Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) / v)));
	elseif (u <= 45.0)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.9e-115)
		tmp = t1 / ((t1 + u) * (-u / v));
	elseif (u <= 45.0)
		tmp = -v / t1;
	else
		tmp = (-t1 / (u / v)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.9e-115], N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 45.0], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.8999999999999998e-115

   1. Initial program 70.4%

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

      1. associate-*l/ 77.7%

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

      2. *-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. associate-/r* 91.8%

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

      2. associate-*r/ 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-/r/ 95.9%

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

      5. div-inv 95.7%

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

      6. frac-2neg 95.7%

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

      7. frac-times 91.3%

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

      8. remove-double-neg 91.3%

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

      9. *-commutative 91.3%

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

      10. *-un-lft-identity 91.3%

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

      11. distribute-neg-frac 91.3%

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

      12. distribute-neg-in 91.3%

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

      13. add-sqr-sqrt 43.2%

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

      14. sqrt-unprod 83.6%

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

      15. sqr-neg 83.6%

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

      16. sqrt-unprod 44.6%

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

      17. add-sqr-sqrt 77.4%

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

      18. sub-neg 77.4%

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

   5. Applied egg-rr 77.4%

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

   6. Taylor expanded in t1 around 0 77.9%

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

      1. neg-mul-1 77.9%

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

      2. distribute-neg-frac 77.9%

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

   8. Simplified 77.9%

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

   if -2.8999999999999998e-115 < u < 45

   1. Initial program 72.0%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in t1 around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

   if 45 < u

   1. Initial program 81.9%

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

      1. associate-/r* 88.3%

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

      2. associate-/l* 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. associate-/l* 84.6%

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

      3. distribute-neg-frac 84.6%

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

   6. Simplified 84.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq 45:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \end{array} \]

Alternative 12: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+85} \lor \neg \left(u \leq 3 \cdot 10^{+101}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.8e+85) (not (<= u 3e+101)))
   (* v (/ t1 (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.8e+85) || !(u <= 3e+101)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.8d+85)) .or. (.not. (u <= 3d+101))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.8e+85) || !(u <= 3e+101)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.8e+85) or not (u <= 3e+101):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.8e+85) || !(u <= 3e+101))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.8e+85) || ~((u <= 3e+101)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.8e+85], N[Not[LessEqual[u, 3e+101]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.79999999999999993e85 or 2.99999999999999993e101 < u

   1. Initial program 74.0%

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

   2. Taylor expanded in t1 around 0 73.0%

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

      1. unpow2 73.0%

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

   4. Simplified 73.0%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 72.1%

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

      3. add-sqr-sqrt 33.4%

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

      4. sqrt-unprod 48.6%

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

      5. sqr-neg 48.6%

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

      6. sqrt-unprod 34.4%

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

      7. add-sqr-sqrt 66.0%

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

   6. Applied egg-rr 66.0%

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

   if -4.79999999999999993e85 < u < 2.99999999999999993e101

   1. Initial program 73.6%

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

      1. associate-*l/ 81.2%

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

      2. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in t1 around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   6. Simplified 68.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+85} \lor \neg \left(u \leq 3 \cdot 10^{+101}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13: 24.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.45 \cdot 10^{+55}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.45e+55) (/ v t1) (if (<= t1 5.6e+89) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.45e+55) {
		tmp = v / t1;
	} else if (t1 <= 5.6e+89) {
		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 (t1 <= (-2.45d+55)) then
        tmp = v / t1
    else if (t1 <= 5.6d+89) 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 (t1 <= -2.45e+55) {
		tmp = v / t1;
	} else if (t1 <= 5.6e+89) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.45e+55:
		tmp = v / t1
	elif t1 <= 5.6e+89:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.45e+55)
		tmp = Float64(v / t1);
	elseif (t1 <= 5.6e+89)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.45e+55)
		tmp = v / t1;
	elseif (t1 <= 5.6e+89)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.45e+55], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 5.6e+89], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.45000000000000007e55 or 5.5999999999999996e89 < t1

   1. Initial program 63.0%

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

      1. associate-*l/ 67.0%

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

      2. *-commutative 67.0%

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

   3. Simplified 67.0%

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

      1. associate-/r* 96.8%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r/ 98.8%

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

      5. div-inv 98.7%

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

      6. frac-2neg 98.7%

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

      7. frac-times 74.4%

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

      8. remove-double-neg 74.4%

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

      9. *-commutative 74.4%

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

      10. *-un-lft-identity 74.4%

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

      11. distribute-neg-frac 74.4%

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

      12. distribute-neg-in 74.4%

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

      13. add-sqr-sqrt 39.5%

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

      14. sqrt-unprod 55.6%

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

      15. sqr-neg 55.6%

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

      16. sqrt-unprod 22.2%

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

      17. add-sqr-sqrt 48.0%

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

      18. sub-neg 48.0%

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

   5. Applied egg-rr 48.0%

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

   6. Taylor expanded in t1 around inf 45.1%

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

   7. Taylor expanded in t1 around inf 40.4%

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

   if -2.45000000000000007e55 < t1 < 5.5999999999999996e89

   1. Initial program 79.9%

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

      1. associate-*l/ 84.9%

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

      2. *-commutative 84.9%

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

   3. Simplified 84.9%

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

      1. associate-/r* 92.4%

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

      2. associate-*r/ 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-/r/ 97.2%

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

      5. div-inv 97.1%

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

      6. frac-2neg 97.1%

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

      7. frac-times 90.5%

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

      8. remove-double-neg 90.5%

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

      9. *-commutative 90.5%

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

      10. *-un-lft-identity 90.5%

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

      11. distribute-neg-frac 90.5%

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

      12. distribute-neg-in 90.5%

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

      13. add-sqr-sqrt 42.0%

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

      14. sqrt-unprod 72.3%

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

      15. sqr-neg 72.3%

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

      16. sqrt-unprod 31.9%

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

      17. add-sqr-sqrt 62.0%

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

      18. sub-neg 62.0%

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

   5. Applied egg-rr 62.0%

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

   6. Taylor expanded in t1 around inf 12.9%

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

   7. Taylor expanded in t1 around 0 15.2%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.45 \cdot 10^{+55}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 14: 56.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 2.35 \cdot 10^{+110}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 2.35e+110) (- (/ v t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.35e+110) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 2.35d+110) then
        tmp = -(v / t1)
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.35e+110) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 2.35e+110:
		tmp = -(v / t1)
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 2.35e+110)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 2.35e+110)
		tmp = -(v / t1);
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 2.35e+110], (-N[(v / t1), $MachinePrecision]), N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 2.3499999999999999e110

   1. Initial program 72.9%

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

      1. associate-*l/ 79.3%

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

      2. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in t1 around inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. neg-mul-1 59.4%

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

   6. Simplified 59.4%

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

   if 2.3499999999999999e110 < u

   1. Initial program 78.4%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. associate-/r* 80.5%

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

      2. associate-*r/ 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-/r/ 98.6%

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

      5. div-inv 98.7%

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

      6. frac-2neg 98.7%

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

      7. frac-times 87.0%

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

      8. remove-double-neg 87.0%

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

      9. *-commutative 87.0%

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

      10. *-un-lft-identity 87.0%

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

      11. distribute-neg-frac 87.0%

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

      12. distribute-neg-in 87.0%

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

      13. add-sqr-sqrt 51.6%

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

      14. sqrt-unprod 87.0%

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

      15. sqr-neg 87.0%

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

      16. sqrt-unprod 35.4%

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

      17. add-sqr-sqrt 87.0%

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

      18. sub-neg 87.0%

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

   5. Applied egg-rr 87.0%

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

   6. Taylor expanded in t1 around inf 49.1%

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

   7. Taylor expanded in t1 around 0 44.1%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 2.35 \cdot 10^{+110}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 15: 62.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in t1 around inf 59.6%

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

   1. neg-mul-1 59.6%

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

6. Simplified 59.6%

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

7. Final simplification 59.6%

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

Alternative 16: 14.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-*l/ 78.4%

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

   2. *-commutative 78.4%

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

3. Simplified 78.4%

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

   1. associate-/r* 94.0%

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

   2. associate-*r/ 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-/r/ 97.8%

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

   5. div-inv 97.6%

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

   6. frac-2neg 97.6%

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

   7. frac-times 84.7%

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

   8. remove-double-neg 84.7%

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

   9. *-commutative 84.7%

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

   10. *-un-lft-identity 84.7%

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

   11. distribute-neg-frac 84.7%

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

   12. distribute-neg-in 84.7%

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

   13. add-sqr-sqrt 41.1%

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

   14. sqrt-unprod 66.2%

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

   15. sqr-neg 66.2%

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

   16. sqrt-unprod 28.3%

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

   17. add-sqr-sqrt 56.9%

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

   18. sub-neg 56.9%

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

5. Applied egg-rr 56.9%

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

6. Taylor expanded in t1 around inf 24.6%

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

7. Taylor expanded in t1 around inf 16.3%

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

8. Final simplification 16.3%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))