Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.3%

Time: 9.4s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-/l* 75.9%

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

   2. distribute-lft-neg-out 75.9%

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

   3. distribute-rgt-neg-in 75.9%

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

   4. associate-/r* 85.9%

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

   5. distribute-neg-frac2 85.9%

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

3. Simplified 85.9%

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

   1. associate-*r/ 98.8%

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

   2. +-commutative 98.8%

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

   3. distribute-neg-in 98.8%

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

   4. sub-neg 98.8%

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

   5. associate-*l/ 98.4%

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

   6. frac-2neg 98.4%

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

   7. associate-*r/ 98.8%

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

   8. remove-double-neg 98.8%

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

   9. sub-neg 98.8%

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

   10. distribute-neg-in 98.8%

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

   11. +-commutative 98.8%

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

   12. frac-2neg 98.8%

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

   13. add-sqr-sqrt 47.8%

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

   14. sqrt-unprod 44.7%

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

   15. sqr-neg 44.7%

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

   16. sqrt-unprod 21.1%

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

   17. add-sqr-sqrt 36.2%

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

   18. add-sqr-sqrt 14.9%

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

   19. sqrt-unprod 60.3%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ t_2 := \frac{v}{t\_1}\\ t_3 := t1 \cdot \frac{t\_2}{t1 + u}\\ \mathbf{if}\;t1 \leq -5.7 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 1.86 \cdot 10^{-196}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+49}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)) (t_2 (/ v t_1)) (t_3 (* t1 (/ t_2 (+ t1 u)))))
   (if (<= t1 -5.7e+144)
     t_2
     (if (<= t1 1.86e-196)
       t_3
       (if (<= t1 9e+49)
         (* v (/ t1 (* (+ t1 u) t_1)))
         (if (<= t1 2.5e+132) t_3 (* (/ t1 (+ t1 u)) (/ v (- t1)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = v / t_1;
	double t_3 = t1 * (t_2 / (t1 + u));
	double tmp;
	if (t1 <= -5.7e+144) {
		tmp = t_2;
	} else if (t1 <= 1.86e-196) {
		tmp = t_3;
	} else if (t1 <= 9e+49) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else if (t1 <= 2.5e+132) {
		tmp = t_3;
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = -t1 - u
    t_2 = v / t_1
    t_3 = t1 * (t_2 / (t1 + u))
    if (t1 <= (-5.7d+144)) then
        tmp = t_2
    else if (t1 <= 1.86d-196) then
        tmp = t_3
    else if (t1 <= 9d+49) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else if (t1 <= 2.5d+132) then
        tmp = t_3
    else
        tmp = (t1 / (t1 + u)) * (v / -t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = v / t_1;
	double t_3 = t1 * (t_2 / (t1 + u));
	double tmp;
	if (t1 <= -5.7e+144) {
		tmp = t_2;
	} else if (t1 <= 1.86e-196) {
		tmp = t_3;
	} else if (t1 <= 9e+49) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else if (t1 <= 2.5e+132) {
		tmp = t_3;
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 - u
	t_2 = v / t_1
	t_3 = t1 * (t_2 / (t1 + u))
	tmp = 0
	if t1 <= -5.7e+144:
		tmp = t_2
	elif t1 <= 1.86e-196:
		tmp = t_3
	elif t1 <= 9e+49:
		tmp = v * (t1 / ((t1 + u) * t_1))
	elif t1 <= 2.5e+132:
		tmp = t_3
	else:
		tmp = (t1 / (t1 + u)) * (v / -t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	t_2 = Float64(v / t_1)
	t_3 = Float64(t1 * Float64(t_2 / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -5.7e+144)
		tmp = t_2;
	elseif (t1 <= 1.86e-196)
		tmp = t_3;
	elseif (t1 <= 9e+49)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	elseif (t1 <= 2.5e+132)
		tmp = t_3;
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 - u;
	t_2 = v / t_1;
	t_3 = t1 * (t_2 / (t1 + u));
	tmp = 0.0;
	if (t1 <= -5.7e+144)
		tmp = t_2;
	elseif (t1 <= 1.86e-196)
		tmp = t_3;
	elseif (t1 <= 9e+49)
		tmp = v * (t1 / ((t1 + u) * t_1));
	elseif (t1 <= 2.5e+132)
		tmp = t_3;
	else
		tmp = (t1 / (t1 + u)) * (v / -t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, Block[{t$95$2 = N[(v / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t1 * N[(t$95$2 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.7e+144], t$95$2, If[LessEqual[t1, 1.86e-196], t$95$3, If[LessEqual[t1, 9e+49], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.5e+132], t$95$3, N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -5.70000000000000005e144

   1. Initial program 39.2%

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

      1. associate-/l* 39.9%

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

      2. distribute-lft-neg-out 39.9%

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

      3. distribute-rgt-neg-in 39.9%

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

      4. associate-/r* 68.6%

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

      5. distribute-neg-frac2 68.6%

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

   3. Simplified 68.6%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. remove-double-neg 99.9%

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

      9. sub-neg 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. +-commutative 99.9%

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

      12. frac-2neg 99.9%

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

      13. add-sqr-sqrt 99.4%

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

      14. sqrt-unprod 7.8%

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

      15. sqr-neg 7.8%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 34.1%

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

      18. add-sqr-sqrt 28.1%

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

      19. sqrt-unprod 35.1%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 92.2%

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

      1. mul-1-neg 92.2%

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

   9. Simplified 92.2%

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

   if -5.70000000000000005e144 < t1 < 1.8600000000000001e-196 or 8.99999999999999965e49 < t1 < 2.5000000000000001e132

   1. Initial program 79.3%

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

      1. associate-/l* 85.3%

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

      2. distribute-lft-neg-out 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

      4. associate-/r* 92.0%

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

      5. distribute-neg-frac2 92.0%

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

   3. Simplified 92.0%

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

   if 1.8600000000000001e-196 < t1 < 8.99999999999999965e49

   1. Initial program 93.8%

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

      1. associate-*l/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 2.5000000000000001e132 < t1

   1. Initial program 52.4%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 88.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 1.86 \cdot 10^{-196}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-t1\right) - u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+49}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-t1\right) - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{-t1}\\ t_2 := \frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{if}\;u \leq -3.4 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq -9.5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -72000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq 1.68 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- t1))) (t_2 (/ t1 (* u (/ u (- v))))))
   (if (<= u -3.4e+62)
     t_2
     (if (<= u -9.5e+36)
       t_1
       (if (<= u -72000000000.0)
         t_2
         (if (<= u 1.68e+16) t_1 (* (/ t1 (- u)) (/ v u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / -t1;
	double t_2 = t1 / (u * (u / -v));
	double tmp;
	if (u <= -3.4e+62) {
		tmp = t_2;
	} else if (u <= -9.5e+36) {
		tmp = t_1;
	} else if (u <= -72000000000.0) {
		tmp = t_2;
	} else if (u <= 1.68e+16) {
		tmp = t_1;
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v / -t1
    t_2 = t1 / (u * (u / -v))
    if (u <= (-3.4d+62)) then
        tmp = t_2
    else if (u <= (-9.5d+36)) then
        tmp = t_1
    else if (u <= (-72000000000.0d0)) then
        tmp = t_2
    else if (u <= 1.68d+16) then
        tmp = t_1
    else
        tmp = (t1 / -u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / -t1;
	double t_2 = t1 / (u * (u / -v));
	double tmp;
	if (u <= -3.4e+62) {
		tmp = t_2;
	} else if (u <= -9.5e+36) {
		tmp = t_1;
	} else if (u <= -72000000000.0) {
		tmp = t_2;
	} else if (u <= 1.68e+16) {
		tmp = t_1;
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / -t1
	t_2 = t1 / (u * (u / -v))
	tmp = 0
	if u <= -3.4e+62:
		tmp = t_2
	elif u <= -9.5e+36:
		tmp = t_1
	elif u <= -72000000000.0:
		tmp = t_2
	elif u <= 1.68e+16:
		tmp = t_1
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(-t1))
	t_2 = Float64(t1 / Float64(u * Float64(u / Float64(-v))))
	tmp = 0.0
	if (u <= -3.4e+62)
		tmp = t_2;
	elseif (u <= -9.5e+36)
		tmp = t_1;
	elseif (u <= -72000000000.0)
		tmp = t_2;
	elseif (u <= 1.68e+16)
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / -t1;
	t_2 = t1 / (u * (u / -v));
	tmp = 0.0;
	if (u <= -3.4e+62)
		tmp = t_2;
	elseif (u <= -9.5e+36)
		tmp = t_1;
	elseif (u <= -72000000000.0)
		tmp = t_2;
	elseif (u <= 1.68e+16)
		tmp = t_1;
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / (-t1)), $MachinePrecision]}, Block[{t$95$2 = N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.4e+62], t$95$2, If[LessEqual[u, -9.5e+36], t$95$1, If[LessEqual[u, -72000000000.0], t$95$2, If[LessEqual[u, 1.68e+16], t$95$1, N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.40000000000000014e62 or -9.49999999999999974e36 < u < -7.2e10

   1. Initial program 77.8%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 82.1%

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

   6. Taylor expanded in t1 around 0 80.6%

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

      1. associate-*r/ 80.6%

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

      2. mul-1-neg 80.6%

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

   8. Simplified 80.6%

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

      1. *-commutative 80.6%

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

      2. clear-num 82.5%

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

      3. frac-2neg 82.5%

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

      4. frac-times 84.4%

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

      5. *-un-lft-identity 84.4%

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

      6. remove-double-neg 84.4%

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

   10. Applied egg-rr 84.4%

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

   if -3.40000000000000014e62 < u < -9.49999999999999974e36 or -7.2e10 < u < 1.68e16

   1. Initial program 68.0%

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

      1. associate-/l* 71.8%

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

      2. distribute-lft-neg-out 71.8%

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

      3. distribute-rgt-neg-in 71.8%

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

      4. associate-/r* 83.5%

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

      5. distribute-neg-frac2 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in t1 around inf 83.3%

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

      1. associate-*r/ 83.3%

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

      2. neg-mul-1 83.3%

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

   7. Simplified 83.3%

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

   if 1.68e16 < u

   1. Initial program 79.8%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around 0 85.9%

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

   6. Taylor expanded in t1 around 0 84.6%

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

      1. associate-*r/ 84.6%

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

      2. mul-1-neg 84.6%

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

   8. Simplified 84.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{elif}\;u \leq -9.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;u \leq -72000000000:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{elif}\;u \leq 1.68 \cdot 10^{+16}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -3.6 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+128}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -3.6e+146)
     t_1
     (if (<= t1 9e+128)
       (* t1 (/ t_1 (+ t1 u)))
       (* (/ t1 (+ t1 u)) (/ v (- t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -3.6e+146) {
		tmp = t_1;
	} else if (t1 <= 9e+128) {
		tmp = t1 * (t_1 / (t1 + u));
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (-t1 - u)
    if (t1 <= (-3.6d+146)) then
        tmp = t_1
    else if (t1 <= 9d+128) then
        tmp = t1 * (t_1 / (t1 + u))
    else
        tmp = (t1 / (t1 + u)) * (v / -t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -3.6e+146) {
		tmp = t_1;
	} else if (t1 <= 9e+128) {
		tmp = t1 * (t_1 / (t1 + u));
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	tmp = 0
	if t1 <= -3.6e+146:
		tmp = t_1
	elif t1 <= 9e+128:
		tmp = t1 * (t_1 / (t1 + u))
	else:
		tmp = (t1 / (t1 + u)) * (v / -t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -3.6e+146)
		tmp = t_1;
	elseif (t1 <= 9e+128)
		tmp = Float64(t1 * Float64(t_1 / Float64(t1 + u)));
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -3.6e+146)
		tmp = t_1;
	elseif (t1 <= 9e+128)
		tmp = t1 * (t_1 / (t1 + u));
	else
		tmp = (t1 / (t1 + u)) * (v / -t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.6e+146], t$95$1, If[LessEqual[t1, 9e+128], N[(t1 * N[(t$95$1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.5999999999999998e146

   1. Initial program 39.2%

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

      1. associate-/l* 39.9%

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

      2. distribute-lft-neg-out 39.9%

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

      3. distribute-rgt-neg-in 39.9%

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

      4. associate-/r* 68.6%

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

      5. distribute-neg-frac2 68.6%

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

   3. Simplified 68.6%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. remove-double-neg 99.9%

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

      9. sub-neg 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. +-commutative 99.9%

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

      12. frac-2neg 99.9%

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

      13. add-sqr-sqrt 99.4%

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

      14. sqrt-unprod 7.8%

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

      15. sqr-neg 7.8%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 34.1%

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

      18. add-sqr-sqrt 28.1%

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

      19. sqrt-unprod 35.1%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 92.2%

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

      1. mul-1-neg 92.2%

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

   9. Simplified 92.2%

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

   if -3.5999999999999998e146 < t1 < 9.0000000000000003e128

   1. Initial program 84.3%

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

      1. associate-/l* 87.8%

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

      2. distribute-lft-neg-out 87.8%

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

      3. distribute-rgt-neg-in 87.8%

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

      4. associate-/r* 92.2%

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

      5. distribute-neg-frac2 92.2%

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

   3. Simplified 92.2%

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

   if 9.0000000000000003e128 < t1

   1. Initial program 52.4%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 88.1%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.6 \cdot 10^{+146}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+128}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-t1\right) - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 360000:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.12e-41)
   (/ v (- (- t1) u))
   (if (<= t1 360000.0)
     (/ (* v (/ t1 u)) (- u))
     (* (/ t1 (+ t1 u)) (/ v (- t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.12e-41) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 360000.0) {
		tmp = (v * (t1 / u)) / -u;
	} else {
		tmp = (t1 / (t1 + u)) * (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 <= (-1.12d-41)) then
        tmp = v / (-t1 - u)
    else if (t1 <= 360000.0d0) then
        tmp = (v * (t1 / u)) / -u
    else
        tmp = (t1 / (t1 + u)) * (v / -t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.12e-41) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 360000.0) {
		tmp = (v * (t1 / u)) / -u;
	} else {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.12e-41:
		tmp = v / (-t1 - u)
	elif t1 <= 360000.0:
		tmp = (v * (t1 / u)) / -u
	else:
		tmp = (t1 / (t1 + u)) * (v / -t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.12e-41)
		tmp = Float64(v / Float64(Float64(-t1) - u));
	elseif (t1 <= 360000.0)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.12e-41)
		tmp = v / (-t1 - u);
	elseif (t1 <= 360000.0)
		tmp = (v * (t1 / u)) / -u;
	else
		tmp = (t1 / (t1 + u)) * (v / -t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.12e-41], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 360000.0], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.11999999999999999e-41

   1. Initial program 60.2%

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

      1. associate-/l* 65.8%

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

      2. distribute-lft-neg-out 65.8%

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

      3. distribute-rgt-neg-in 65.8%

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

      4. associate-/r* 81.2%

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

      5. distribute-neg-frac2 81.2%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. remove-double-neg 99.9%

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

      9. sub-neg 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. +-commutative 99.9%

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

      12. frac-2neg 99.9%

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

      13. add-sqr-sqrt 99.4%

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

      14. sqrt-unprod 58.6%

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

      15. sqr-neg 58.6%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 26.9%

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

      18. add-sqr-sqrt 21.4%

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

      19. sqrt-unprod 29.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 80.7%

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

      1. mul-1-neg 80.7%

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

   9. Simplified 80.7%

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

   if -1.11999999999999999e-41 < t1 < 3.6e5

   1. Initial program 87.9%

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

      1. times-frac 96.7%

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

      2. distribute-frac-neg 96.7%

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

      3. distribute-neg-frac2 96.7%

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

      4. +-commutative 96.7%

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

      5. distribute-neg-in 96.7%

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

      6. unsub-neg 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in t1 around 0 74.1%

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

   6. Taylor expanded in t1 around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. mul-1-neg 77.9%

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

   8. Simplified 77.9%

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

      1. associate-*l/ 78.6%

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

      2. *-commutative 78.6%

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

      3. frac-2neg 78.6%

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

      4. distribute-rgt-neg-in 78.6%

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

      5. remove-double-neg 78.6%

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

      6. add-sqr-sqrt 45.2%

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

      7. sqrt-unprod 52.2%

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

      8. sqr-neg 52.2%

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

      9. sqrt-unprod 15.1%

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

      10. add-sqr-sqrt 40.4%

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

      11. associate-*l/ 40.6%

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

      12. associate-/l* 40.4%

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

      13. add-sqr-sqrt 15.1%

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

      14. sqrt-unprod 53.1%

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

      15. sqr-neg 53.1%

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

      16. sqrt-unprod 45.2%

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

      17. add-sqr-sqrt 78.7%

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

   10. Applied egg-rr 78.7%

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

   if 3.6e5 < t1

   1. Initial program 62.6%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 85.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 360000:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8.6 \cdot 10^{-43} \lor \neg \left(t1 \leq 280000\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -8.6e-43) (not (<= t1 280000.0)))
   (/ v (- (- t1) u))
   (/ (* v (/ t1 u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8.6e-43) || !(t1 <= 280000.0)) {
		tmp = v / (-t1 - u);
	} 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) :: tmp
    if ((t1 <= (-8.6d-43)) .or. (.not. (t1 <= 280000.0d0))) then
        tmp = v / (-t1 - u)
    else
        tmp = (v * (t1 / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8.6e-43) || !(t1 <= 280000.0)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -8.6e-43) or not (t1 <= 280000.0):
		tmp = v / (-t1 - u)
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -8.6e-43) || !(t1 <= 280000.0))
		tmp = Float64(v / Float64(Float64(-t1) - u));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -8.6e-43) || ~((t1 <= 280000.0)))
		tmp = v / (-t1 - u);
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -8.6e-43], N[Not[LessEqual[t1, 280000.0]], $MachinePrecision]], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.59999999999999927e-43 or 2.8e5 < t1

   1. Initial program 61.3%

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

      1. associate-/l* 65.3%

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

      2. distribute-lft-neg-out 65.3%

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

      3. distribute-rgt-neg-in 65.3%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. remove-double-neg 99.9%

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

      9. sub-neg 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. +-commutative 99.9%

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

      12. frac-2neg 99.9%

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

      13. add-sqr-sqrt 55.0%

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

      14. sqrt-unprod 39.4%

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

      15. sqr-neg 39.4%

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

      16. sqrt-unprod 17.7%

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

      17. add-sqr-sqrt 32.6%

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

      18. add-sqr-sqrt 15.4%

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

      19. sqrt-unprod 46.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 82.7%

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

      1. mul-1-neg 82.7%

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

   9. Simplified 82.7%

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

   if -8.59999999999999927e-43 < t1 < 2.8e5

   1. Initial program 87.9%

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

      1. times-frac 96.7%

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

      2. distribute-frac-neg 96.7%

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

      3. distribute-neg-frac2 96.7%

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

      4. +-commutative 96.7%

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

      5. distribute-neg-in 96.7%

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

      6. unsub-neg 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in t1 around 0 74.1%

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

   6. Taylor expanded in t1 around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. mul-1-neg 77.9%

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

   8. Simplified 77.9%

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

      1. associate-*l/ 78.6%

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

      2. *-commutative 78.6%

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

      3. frac-2neg 78.6%

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

      4. distribute-rgt-neg-in 78.6%

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

      5. remove-double-neg 78.6%

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

      6. add-sqr-sqrt 45.2%

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

      7. sqrt-unprod 52.2%

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

      8. sqr-neg 52.2%

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

      9. sqrt-unprod 15.1%

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

      10. add-sqr-sqrt 40.4%

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

      11. associate-*l/ 40.6%

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

      12. associate-/l* 40.4%

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

      13. add-sqr-sqrt 15.1%

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

      14. sqrt-unprod 53.1%

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

      15. sqr-neg 53.1%

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

      16. sqrt-unprod 45.2%

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

      17. add-sqr-sqrt 78.7%

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

   10. Applied egg-rr 78.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.6 \cdot 10^{-43} \lor \neg \left(t1 \leq 280000\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+63} \lor \neg \left(u \leq 15000000000000\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.4e+63) (not (<= u 15000000000000.0)))
   (* (/ t1 (- u)) (/ v u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.4e+63) || !(u <= 15000000000000.0)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6.4d+63)) .or. (.not. (u <= 15000000000000.0d0))) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.4e+63) || !(u <= 15000000000000.0)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.4e+63) or not (u <= 15000000000000.0):
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.4e+63) || !(u <= 15000000000000.0))
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.4e+63) || ~((u <= 15000000000000.0)))
		tmp = (t1 / -u) * (v / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.4e+63], N[Not[LessEqual[u, 15000000000000.0]], $MachinePrecision]], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.40000000000000022e63 or 1.5e13 < u

   1. Initial program 77.5%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 84.2%

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

   6. Taylor expanded in t1 around 0 83.4%

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

      1. associate-*r/ 83.4%

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

      2. mul-1-neg 83.4%

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

   8. Simplified 83.4%

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

   if -6.40000000000000022e63 < u < 1.5e13

   1. Initial program 69.8%

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

      1. associate-/l* 73.4%

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

      2. distribute-lft-neg-out 73.4%

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

      3. distribute-rgt-neg-in 73.4%

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

      4. associate-/r* 84.4%

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

      5. distribute-neg-frac2 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in t1 around inf 80.1%

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

      1. associate-*r/ 80.1%

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

      2. neg-mul-1 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+63} \lor \neg \left(u \leq 15000000000000\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+202} \lor \neg \left(u \leq 1.2 \cdot 10^{+22}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.3e+202) (not (<= u 1.2e+22)))
   (* t1 (/ (/ v u) u))
   (/ v (- (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e+202) || !(u <= 1.2e+22)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / (-t1 - u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.3d+202)) .or. (.not. (u <= 1.2d+22))) then
        tmp = t1 * ((v / u) / u)
    else
        tmp = v / (-t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e+202) || !(u <= 1.2e+22)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / (-t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.3e+202) or not (u <= 1.2e+22):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = v / (-t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.3e+202) || !(u <= 1.2e+22))
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	else
		tmp = Float64(v / Float64(Float64(-t1) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.3e+202) || ~((u <= 1.2e+22)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = v / (-t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.3e+202], N[Not[LessEqual[u, 1.2e+22]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.3000000000000001e202 or 1.2e22 < u

   1. Initial program 81.2%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 87.5%

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

   6. Taylor expanded in t1 around 0 86.4%

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

      1. associate-*r/ 86.4%

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

      2. mul-1-neg 86.4%

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

   8. Simplified 86.4%

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

      1. *-commutative 86.4%

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

      2. clear-num 86.2%

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

      3. un-div-inv 86.2%

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

      4. add-sqr-sqrt 36.8%

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

      5. sqrt-unprod 58.9%

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

      6. sqr-neg 58.9%

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

      7. sqrt-unprod 36.7%

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

      8. add-sqr-sqrt 65.0%

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

   10. Applied egg-rr 65.0%

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

       1. associate-/r/ 67.2%

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

   12. Applied egg-rr 67.2%

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

   if -1.3000000000000001e202 < u < 1.2e22

   1. Initial program 68.9%

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

      1. associate-/l* 72.6%

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

      2. distribute-lft-neg-out 72.6%

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

      3. distribute-rgt-neg-in 72.6%

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

      4. associate-/r* 84.3%

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

      5. distribute-neg-frac2 84.3%

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

   3. Simplified 84.3%

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

      1. associate-*r/ 98.2%

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

      2. +-commutative 98.2%

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

      3. distribute-neg-in 98.2%

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

      4. sub-neg 98.2%

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

      5. associate-*l/ 98.2%

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

      6. frac-2neg 98.2%

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

      7. associate-*r/ 98.7%

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

      8. remove-double-neg 98.7%

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

      9. sub-neg 98.7%

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

      10. distribute-neg-in 98.7%

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

      11. +-commutative 98.7%

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

      12. frac-2neg 98.7%

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

      13. add-sqr-sqrt 49.6%

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

      14. sqrt-unprod 36.2%

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

      15. sqr-neg 36.2%

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

      16. sqrt-unprod 11.5%

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

      17. add-sqr-sqrt 18.8%

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

      18. add-sqr-sqrt 11.5%

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

      19. sqrt-unprod 49.2%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in t1 around inf 75.6%

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

      1. mul-1-neg 75.6%

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

   9. Simplified 75.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+202} \lor \neg \left(u \leq 1.2 \cdot 10^{+22}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+182} \lor \neg \left(u \leq 6.7 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1e+182) (not (<= u 6.7e+151))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1e+182) || !(u <= 6.7e+151)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1d+182)) .or. (.not. (u <= 6.7d+151))) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1e+182) || !(u <= 6.7e+151)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1e+182) or not (u <= 6.7e+151):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1e+182) || !(u <= 6.7e+151))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1e+182) || ~((u <= 6.7e+151)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1e+182], N[Not[LessEqual[u, 6.7e+151]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.0000000000000001e182 or 6.69999999999999964e151 < u

   1. Initial program 77.4%

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

      1. associate-/l* 78.2%

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

      2. distribute-lft-neg-out 78.2%

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

      3. distribute-rgt-neg-in 78.2%

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

      4. associate-/r* 90.9%

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

      5. distribute-neg-frac2 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in t1 around inf 44.9%

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

   6. Taylor expanded in t1 around 0 39.1%

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

      1. associate-*r/ 39.1%

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

      2. mul-1-neg 39.1%

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

   8. Simplified 39.1%

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

   if -1.0000000000000001e182 < u < 6.69999999999999964e151

   1. Initial program 71.9%

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

      1. associate-/l* 75.2%

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

      2. distribute-lft-neg-out 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. associate-/r* 84.4%

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

      5. distribute-neg-frac2 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in t1 around inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. neg-mul-1 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+182} \lor \neg \left(u \leq 6.7 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+178} \lor \neg \left(u \leq 4.4 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.8e+178) (not (<= u 4.4e+151))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e+178) || !(u <= 4.4e+151)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.8d+178)) .or. (.not. (u <= 4.4d+151))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e+178) || !(u <= 4.4e+151)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.8e+178) or not (u <= 4.4e+151):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.8e+178) || !(u <= 4.4e+151))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.8e+178) || ~((u <= 4.4e+151)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.8e+178], N[Not[LessEqual[u, 4.4e+151]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.8000000000000001e178 or 4.40000000000000013e151 < u

   1. Initial program 77.4%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. clear-num 93.9%

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

      3. frac-2neg 93.9%

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

      4. frac-times 86.5%

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

      5. *-un-lft-identity 86.5%

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

      6. sub-neg 86.5%

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

      7. distribute-neg-in 86.5%

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

      8. +-commutative 86.5%

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

      9. remove-double-neg 86.5%

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

   7. Applied egg-rr 86.5%

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

      1. frac-2neg 86.5%

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

      2. div-inv 86.5%

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

      3. remove-double-neg 86.5%

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

      4. distribute-rgt-neg-in 86.5%

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

      5. distribute-neg-in 86.5%

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

      6. add-sqr-sqrt 39.0%

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

      7. sqrt-unprod 83.1%

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

      8. sqr-neg 83.1%

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

      9. sqrt-unprod 47.6%

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

      10. add-sqr-sqrt 86.5%

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

      11. sub-neg 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. associate-*r/ 86.5%

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

       2. times-frac 94.0%

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

       3. associate-/r/ 94.0%

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

       4. *-commutative 94.0%

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

       5. associate-*r/ 94.0%

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

       6. *-rgt-identity 94.0%

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

       7. associate-*r/ 81.1%

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

       8. *-commutative 81.1%

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

       9. associate-/l* 94.0%

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

   11. Simplified 94.0%

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

   12. Taylor expanded in t1 around inf 38.7%

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

   if -5.8000000000000001e178 < u < 4.40000000000000013e151

   1. Initial program 71.9%

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

      1. associate-/l* 75.2%

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

      2. distribute-lft-neg-out 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. associate-/r* 84.4%

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

      5. distribute-neg-frac2 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in t1 around inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. neg-mul-1 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+178} \lor \neg \left(u \leq 4.4 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+62} \lor \neg \left(t1 \leq 1.08 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.2e+62) (not (<= t1 1.08e+80))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.2e+62) || !(t1 <= 1.08e+80)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-7.2d+62)) .or. (.not. (t1 <= 1.08d+80))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.2e+62) || !(t1 <= 1.08e+80)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.2e+62) or not (t1 <= 1.08e+80):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.2e+62) || !(t1 <= 1.08e+80))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7.2e+62) || ~((t1 <= 1.08e+80)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.2e+62], N[Not[LessEqual[t1, 1.08e+80]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.2e62 or 1.08e80 < t1

   1. Initial program 50.4%

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

      1. associate-*l/ 57.9%

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

      2. *-commutative 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in t1 around inf 86.8%

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

      1. frac-2neg 86.8%

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

      2. metadata-eval 86.8%

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

      3. un-div-inv 87.1%

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

      4. add-sqr-sqrt 46.3%

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

      5. sqrt-unprod 49.4%

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

      6. sqr-neg 49.4%

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

      7. sqrt-unprod 18.4%

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

      8. add-sqr-sqrt 34.4%

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

   7. Applied egg-rr 34.4%

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

   if -7.2e62 < t1 < 1.08e80

   1. Initial program 88.4%

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

      1. times-frac 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-neg-frac2 97.5%

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

      4. +-commutative 97.5%

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

      5. distribute-neg-in 97.5%

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

      6. unsub-neg 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around 0 66.8%

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

      1. *-commutative 66.8%

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

      2. clear-num 66.8%

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

      3. frac-2neg 66.8%

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

      4. frac-times 65.7%

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

      5. *-un-lft-identity 65.7%

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

      6. sub-neg 65.7%

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

      7. distribute-neg-in 65.7%

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

      8. +-commutative 65.7%

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

      9. remove-double-neg 65.7%

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

   7. Applied egg-rr 65.7%

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

      1. frac-2neg 65.7%

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

      2. div-inv 65.6%

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

      3. remove-double-neg 65.6%

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

      4. distribute-rgt-neg-in 65.6%

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

      5. distribute-neg-in 65.6%

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

      6. add-sqr-sqrt 32.9%

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

      7. sqrt-unprod 68.4%

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

      8. sqr-neg 68.4%

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

      9. sqrt-unprod 34.1%

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

      10. add-sqr-sqrt 64.9%

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

      11. sub-neg 64.9%

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

   9. Applied egg-rr 64.9%

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

       1. associate-*r/ 65.0%

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

       2. times-frac 67.0%

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

       3. associate-/r/ 66.6%

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

       4. *-commutative 66.6%

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

       5. associate-*r/ 66.6%

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

       6. *-rgt-identity 66.6%

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

       7. associate-*r/ 65.2%

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

       8. *-commutative 65.2%

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

       9. associate-/l* 66.6%

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

   11. Simplified 66.6%

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

   12. Taylor expanded in t1 around inf 17.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+62} \lor \neg \left(t1 \leq 1.08 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.9% accurate, 1.0× speedup

Math

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

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(v / Float64(t1 + u))) / Float64(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 73.3%

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

   1. associate-/l* 75.9%

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

   2. distribute-lft-neg-out 75.9%

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

   3. distribute-rgt-neg-in 75.9%

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

   4. associate-/r* 85.9%

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

   5. distribute-neg-frac2 85.9%

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

3. Simplified 85.9%

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

   1. associate-*r/ 98.8%

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

   2. frac-2neg 98.8%

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

   3. remove-double-neg 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 13: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 14: 60.9% accurate, 2.0× speedup

Math

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

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(v / Float64(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 73.3%

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

   1. associate-/l* 75.9%

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

   2. distribute-lft-neg-out 75.9%

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

   3. distribute-rgt-neg-in 75.9%

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

   4. associate-/r* 85.9%

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

   5. distribute-neg-frac2 85.9%

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

3. Simplified 85.9%

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

   1. associate-*r/ 98.8%

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

   2. +-commutative 98.8%

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

   3. distribute-neg-in 98.8%

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

   4. sub-neg 98.8%

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

   5. associate-*l/ 98.4%

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

   6. frac-2neg 98.4%

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

   7. associate-*r/ 98.8%

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

   8. remove-double-neg 98.8%

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

   9. sub-neg 98.8%

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

   10. distribute-neg-in 98.8%

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

   11. +-commutative 98.8%

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

   12. frac-2neg 98.8%

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

   13. add-sqr-sqrt 47.8%

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

   14. sqrt-unprod 44.7%

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

   15. sqr-neg 44.7%

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

   16. sqrt-unprod 21.1%

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

   17. add-sqr-sqrt 36.2%

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

   18. add-sqr-sqrt 14.9%

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

   19. sqrt-unprod 60.3%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in t1 around inf 63.4%

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

   1. mul-1-neg 63.4%

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

9. Simplified 63.4%

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

10. Final simplification 63.4%

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

Alternative 15: 14.2% 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 73.3%

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

5. Taylor expanded in t1 around inf 56.1%

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

   1. frac-2neg 56.1%

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

   2. metadata-eval 56.1%

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

   3. un-div-inv 56.2%

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

   4. add-sqr-sqrt 27.7%

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

   5. sqrt-unprod 28.1%

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

   6. sqr-neg 28.1%

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

   7. sqrt-unprod 8.0%

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

   8. add-sqr-sqrt 14.9%

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

7. Applied egg-rr 14.9%

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

Reproduce

herbie shell --seed 2024100 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))