Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.9%

Time: 11.2s

Alternatives: 14

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: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. times-frac 98.9%

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

   2. distribute-frac-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. distribute-neg-in 98.9%

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

   6. unsub-neg 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := t1 \cdot \frac{v}{t\_1 \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -3.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-154}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{+66}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (* t1 (/ v (* t_1 (+ t1 u))))))
   (if (<= t1 -3.8e+124)
     (/ v (- u t1))
     (if (<= t1 -1.65e-154)
       t_2
       (if (<= t1 2.8e-195)
         (* (/ t1 (- u)) (/ v u))
         (if (<= t1 1.75e+66) t_2 (/ v t_1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = t1 * (v / (t_1 * (t1 + u)));
	double tmp;
	if (t1 <= -3.8e+124) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.65e-154) {
		tmp = t_2;
	} else if (t1 <= 2.8e-195) {
		tmp = (t1 / -u) * (v / u);
	} else if (t1 <= 1.75e+66) {
		tmp = t_2;
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = t1 * (v / (t_1 * (t1 + u)))
    if (t1 <= (-3.8d+124)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.65d-154)) then
        tmp = t_2
    else if (t1 <= 2.8d-195) then
        tmp = (t1 / -u) * (v / u)
    else if (t1 <= 1.75d+66) then
        tmp = t_2
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = t1 * (v / (t_1 * (t1 + u)));
	double tmp;
	if (t1 <= -3.8e+124) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.65e-154) {
		tmp = t_2;
	} else if (t1 <= 2.8e-195) {
		tmp = (t1 / -u) * (v / u);
	} else if (t1 <= 1.75e+66) {
		tmp = t_2;
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = t1 * (v / (t_1 * (t1 + u)))
	tmp = 0
	if t1 <= -3.8e+124:
		tmp = v / (u - t1)
	elif t1 <= -1.65e-154:
		tmp = t_2
	elif t1 <= 2.8e-195:
		tmp = (t1 / -u) * (v / u)
	elif t1 <= 1.75e+66:
		tmp = t_2
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(t1 * Float64(v / Float64(t_1 * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3.8e+124)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.65e-154)
		tmp = t_2;
	elseif (t1 <= 2.8e-195)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	elseif (t1 <= 1.75e+66)
		tmp = t_2;
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = t1 * (v / (t_1 * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3.8e+124)
		tmp = v / (u - t1);
	elseif (t1 <= -1.65e-154)
		tmp = t_2;
	elseif (t1 <= 2.8e-195)
		tmp = (t1 / -u) * (v / u);
	elseif (t1 <= 1.75e+66)
		tmp = t_2;
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(t1 * N[(v / N[(t$95$1 * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.8e+124], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.65e-154], t$95$2, If[LessEqual[t1, 2.8e-195], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.75e+66], t$95$2, N[(v / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.7999999999999998e124

   1. Initial program 37.8%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 90.4%

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

      1. mul-1-neg 90.4%

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

      2. frac-2neg 90.4%

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

      3. distribute-neg-frac2 90.4%

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

      4. add-sqr-sqrt 40.0%

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

      5. sqrt-unprod 51.7%

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

      6. sqr-neg 51.7%

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

      7. sqrt-unprod 23.5%

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

      8. add-sqr-sqrt 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. add-sqr-sqrt 34.0%

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

      11. sqrt-unprod 34.8%

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

      12. sqr-neg 34.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 90.5%

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

      15. sub-neg 90.5%

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

   7. Applied egg-rr 90.5%

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

      1. sub-neg 90.5%

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

      2. distribute-neg-in 90.5%

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

      3. remove-double-neg 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in v around 0 90.5%

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

   if -3.7999999999999998e124 < t1 < -1.65000000000000014e-154 or 2.80000000000000003e-195 < t1 < 1.7499999999999999e66

   1. Initial program 89.6%

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

      1. associate-/l* 95.5%

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

   3. Simplified 95.5%

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

   if -1.65000000000000014e-154 < t1 < 2.80000000000000003e-195

   1. Initial program 63.5%

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

      1. times-frac 96.0%

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

      2. distribute-frac-neg 96.0%

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

      3. distribute-neg-frac2 96.0%

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

      4. +-commutative 96.0%

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

      5. distribute-neg-in 96.0%

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

      6. unsub-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t1 around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. mul-1-neg 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in t1 around 0 86.0%

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

   if 1.7499999999999999e66 < t1

   1. Initial program 70.3%

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

      1. associate-*l/ 72.9%

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

      2. *-commutative 72.9%

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

   3. Simplified 72.9%

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

      1. associate-*r/ 70.3%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 10.7%

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

      11. sqr-neg 10.7%

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

      12. sqrt-unprod 45.4%

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

      13. add-sqr-sqrt 45.4%

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

      14. sub-neg 45.4%

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

      15. +-commutative 45.4%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 73.1%

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

      18. sqr-neg 73.1%

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

      19. sqrt-unprod 97.6%

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

      20. add-sqr-sqrt 48.8%

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

      21. sqrt-unprod 97.6%

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

      22. sqr-neg 97.6%

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

      23. sqrt-unprod 50.9%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 97.9%

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

      1. mul-1-neg 97.9%

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

   9. Simplified 97.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-154}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(\left(-u\right) - t1\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{+66}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(\left(-u\right) - t1\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+129}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.6 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 3.3 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -7.2e+129)
     (/ v (- u t1))
     (if (<= t1 -1.6e-140)
       t_1
       (if (<= t1 1.3e-193)
         (* (/ t1 (- u)) (/ v u))
         (if (<= t1 3.3e+66) t_1 (/ v (- (- u) t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -7.2e+129) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.6e-140) {
		tmp = t_1;
	} else if (t1 <= 1.3e-193) {
		tmp = (t1 / -u) * (v / u);
	} else if (t1 <= 3.3e+66) {
		tmp = t_1;
	} else {
		tmp = v / (-u - 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 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-7.2d+129)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.6d-140)) then
        tmp = t_1
    else if (t1 <= 1.3d-193) then
        tmp = (t1 / -u) * (v / u)
    else if (t1 <= 3.3d+66) then
        tmp = t_1
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -7.2e+129) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.6e-140) {
		tmp = t_1;
	} else if (t1 <= 1.3e-193) {
		tmp = (t1 / -u) * (v / u);
	} else if (t1 <= 3.3e+66) {
		tmp = t_1;
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -7.2e+129:
		tmp = v / (u - t1)
	elif t1 <= -1.6e-140:
		tmp = t_1
	elif t1 <= 1.3e-193:
		tmp = (t1 / -u) * (v / u)
	elif t1 <= 3.3e+66:
		tmp = t_1
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -7.2e+129)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.6e-140)
		tmp = t_1;
	elseif (t1 <= 1.3e-193)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	elseif (t1 <= 3.3e+66)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -7.2e+129)
		tmp = v / (u - t1);
	elseif (t1 <= -1.6e-140)
		tmp = t_1;
	elseif (t1 <= 1.3e-193)
		tmp = (t1 / -u) * (v / u);
	elseif (t1 <= 3.3e+66)
		tmp = t_1;
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.2e+129], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.6e-140], t$95$1, If[LessEqual[t1, 1.3e-193], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.3e+66], t$95$1, N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -7.2000000000000002e129

   1. Initial program 37.8%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 90.4%

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

      1. mul-1-neg 90.4%

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

      2. frac-2neg 90.4%

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

      3. distribute-neg-frac2 90.4%

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

      4. add-sqr-sqrt 40.0%

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

      5. sqrt-unprod 51.7%

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

      6. sqr-neg 51.7%

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

      7. sqrt-unprod 23.5%

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

      8. add-sqr-sqrt 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. add-sqr-sqrt 34.0%

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

      11. sqrt-unprod 34.8%

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

      12. sqr-neg 34.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 90.5%

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

      15. sub-neg 90.5%

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

   7. Applied egg-rr 90.5%

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

      1. sub-neg 90.5%

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

      2. distribute-neg-in 90.5%

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

      3. remove-double-neg 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in v around 0 90.5%

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

   if -7.2000000000000002e129 < t1 < -1.6000000000000001e-140 or 1.30000000000000004e-193 < t1 < 3.3000000000000001e66

   1. Initial program 89.2%

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

      1. associate-*l/ 95.1%

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

      2. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   if -1.6000000000000001e-140 < t1 < 1.30000000000000004e-193

   1. Initial program 66.0%

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

      1. times-frac 96.2%

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

      2. distribute-frac-neg 96.2%

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

      3. distribute-neg-frac2 96.2%

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

      4. +-commutative 96.2%

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

      5. distribute-neg-in 96.2%

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

      6. unsub-neg 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t1 around 0 86.9%

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

      1. associate-*r/ 86.9%

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

      2. mul-1-neg 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in t1 around 0 87.0%

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

   if 3.3000000000000001e66 < t1

   1. Initial program 70.3%

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

      1. associate-*l/ 72.9%

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

      2. *-commutative 72.9%

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

   3. Simplified 72.9%

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

      1. associate-*r/ 70.3%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 10.7%

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

      11. sqr-neg 10.7%

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

      12. sqrt-unprod 45.4%

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

      13. add-sqr-sqrt 45.4%

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

      14. sub-neg 45.4%

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

      15. +-commutative 45.4%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 73.1%

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

      18. sqr-neg 73.1%

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

      19. sqrt-unprod 97.6%

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

      20. add-sqr-sqrt 48.8%

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

      21. sqrt-unprod 97.6%

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

      22. sqr-neg 97.6%

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

      23. sqrt-unprod 50.9%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 97.9%

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

      1. mul-1-neg 97.9%

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

   9. Simplified 97.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+129}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.6 \cdot 10^{-140}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 3.3 \cdot 10^{+66}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{-49} \lor \neg \left(u \leq 4.1 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.3e-49) (not (<= u 4.1e-34)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e-49) || !(u <= 4.1e-34)) {
		tmp = (v / (t1 + u)) * (t1 / -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 <= (-1.3d-49)) .or. (.not. (u <= 4.1d-34))) then
        tmp = (v / (t1 + u)) * (t1 / -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 <= -1.3e-49) || !(u <= 4.1e-34)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.3e-49) or not (u <= 4.1e-34):
		tmp = (v / (t1 + u)) * (t1 / -u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.3e-49) || !(u <= 4.1e-34))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / 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 <= -1.3e-49) || ~((u <= 4.1e-34)))
		tmp = (v / (t1 + u)) * (t1 / -u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.3e-49], N[Not[LessEqual[u, 4.1e-34]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.29999999999999997e-49 or 4.1000000000000004e-34 < u

   1. Initial program 81.6%

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

      1. times-frac 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-neg-frac2 99.4%

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

      4. +-commutative 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t1 around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. mul-1-neg 81.9%

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

   7. Simplified 81.9%

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

   if -1.29999999999999997e-49 < u < 4.1000000000000004e-34

   1. Initial program 63.0%

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

      1. associate-*l/ 73.7%

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

      2. *-commutative 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in t1 around inf 80.7%

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

      1. associate-*r/ 80.7%

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

      2. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{-49} \lor \neg \left(u \leq 4.1 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+22} \lor \neg \left(t1 \leq 2.5 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e+22) (not (<= t1 2.5e-119)))
   (/ v (- u t1))
   (* (/ t1 (- u)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+22) || !(t1 <= 2.5e-119)) {
		tmp = v / (u - t1);
	} 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) :: tmp
    if ((t1 <= (-3d+22)) .or. (.not. (t1 <= 2.5d-119))) then
        tmp = v / (u - t1)
    else
        tmp = (t1 / -u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+22) || !(t1 <= 2.5e-119)) {
		tmp = v / (u - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e+22) or not (t1 <= 2.5e-119):
		tmp = v / (u - t1)
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3e+22) || !(t1 <= 2.5e-119))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3e+22) || ~((t1 <= 2.5e-119)))
		tmp = v / (u - t1);
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3e+22], N[Not[LessEqual[t1, 2.5e-119]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3e22 or 2.49999999999999996e-119 < t1

   1. Initial program 68.4%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 83.3%

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

      1. mul-1-neg 83.3%

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

      2. frac-2neg 83.3%

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

      3. distribute-neg-frac2 83.3%

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

      4. add-sqr-sqrt 37.4%

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

      5. sqrt-unprod 41.4%

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

      6. sqr-neg 41.4%

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

      7. sqrt-unprod 16.6%

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

      8. add-sqr-sqrt 28.4%

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

      9. distribute-neg-in 28.4%

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

      10. add-sqr-sqrt 11.7%

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

      11. sqrt-unprod 49.8%

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

      12. sqr-neg 49.8%

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

      13. sqrt-unprod 45.2%

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

      14. add-sqr-sqrt 83.5%

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

      15. sub-neg 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. sub-neg 83.5%

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

      2. distribute-neg-in 83.5%

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

      3. remove-double-neg 83.5%

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

   9. Simplified 83.5%

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

   10. Taylor expanded in v around 0 83.5%

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

   if -3e22 < t1 < 2.49999999999999996e-119

   1. Initial program 77.7%

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

      1. times-frac 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. unsub-neg 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t1 around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. mul-1-neg 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in t1 around 0 77.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+22} \lor \neg \left(t1 \leq 2.5 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+18} \lor \neg \left(t1 \leq 1.6 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.4e+18) (not (<= t1 1.6e-121)))
   (/ v (- u t1))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.4e+18) || !(t1 <= 1.6e-121)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.4d+18)) .or. (.not. (t1 <= 1.6d-121))) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.4e+18) || !(t1 <= 1.6e-121)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.4e+18) or not (t1 <= 1.6e-121):
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.4e+18) || !(t1 <= 1.6e-121))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.4e+18) || ~((t1 <= 1.6e-121)))
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.4e+18], N[Not[LessEqual[t1, 1.6e-121]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4e18 or 1.60000000000000009e-121 < t1

   1. Initial program 68.4%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 83.3%

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

      1. mul-1-neg 83.3%

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

      2. frac-2neg 83.3%

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

      3. distribute-neg-frac2 83.3%

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

      4. add-sqr-sqrt 37.4%

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

      5. sqrt-unprod 41.4%

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

      6. sqr-neg 41.4%

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

      7. sqrt-unprod 16.6%

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

      8. add-sqr-sqrt 28.4%

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

      9. distribute-neg-in 28.4%

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

      10. add-sqr-sqrt 11.7%

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

      11. sqrt-unprod 49.8%

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

      12. sqr-neg 49.8%

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

      13. sqrt-unprod 45.2%

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

      14. add-sqr-sqrt 83.5%

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

      15. sub-neg 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. sub-neg 83.5%

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

      2. distribute-neg-in 83.5%

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

      3. remove-double-neg 83.5%

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

   9. Simplified 83.5%

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

   10. Taylor expanded in v around 0 83.5%

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

   if -1.4e18 < t1 < 1.60000000000000009e-121

   1. Initial program 77.7%

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

      1. associate-/l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t1 around 0 66.8%

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

   6. Taylor expanded in t1 around 0 68.0%

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

      1. associate-/r* 74.0%

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

      2. div-inv 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. *-rgt-identity 74.0%

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

   10. Simplified 74.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+18} \lor \neg \left(t1 \leq 1.6 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-46} \lor \neg \left(u \leq 1.3 \cdot 10^{-33}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5e-46) (not (<= u 1.3e-33)))
   (* t1 (/ v (* u (- u))))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e-46) || !(u <= 1.3e-33)) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5d-46)) .or. (.not. (u <= 1.3d-33))) then
        tmp = t1 * (v / (u * -u))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e-46) || !(u <= 1.3e-33)) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5e-46) or not (u <= 1.3e-33):
		tmp = t1 * (v / (u * -u))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5e-46) || !(u <= 1.3e-33))
		tmp = Float64(t1 * Float64(v / Float64(u * 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 <= -5e-46) || ~((u <= 1.3e-33)))
		tmp = t1 * (v / (u * -u));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5e-46], N[Not[LessEqual[u, 1.3e-33]], $MachinePrecision]], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.99999999999999992e-46 or 1.29999999999999997e-33 < u

   1. Initial program 81.6%

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

      1. associate-/l* 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in t1 around 0 76.7%

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

   6. Taylor expanded in t1 around 0 73.3%

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

   if -4.99999999999999992e-46 < u < 1.29999999999999997e-33

   1. Initial program 63.0%

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

      1. associate-*l/ 73.7%

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

      2. *-commutative 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in t1 around inf 80.7%

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

      1. associate-*r/ 80.7%

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

      2. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-46} \lor \neg \left(u \leq 1.3 \cdot 10^{-33}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+181} \lor \neg \left(u \leq 6.8 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.65e+181) (not (<= u 6.8e+202)))
   (/ v (* u (/ u t1)))
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.65e+181) || !(u <= 6.8e+202)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.65d+181)) .or. (.not. (u <= 6.8d+202))) then
        tmp = v / (u * (u / t1))
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.65e+181) || !(u <= 6.8e+202)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.65e+181) or not (u <= 6.8e+202):
		tmp = v / (u * (u / t1))
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.65e+181) || !(u <= 6.8e+202))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.65e+181) || ~((u <= 6.8e+202)))
		tmp = v / (u * (u / t1));
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.65e+181], N[Not[LessEqual[u, 6.8e+202]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.65000000000000008e181 or 6.8e202 < u

   1. Initial program 84.2%

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

      1. times-frac 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t1 around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. mul-1-neg 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in t1 around 0 95.8%

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

      1. clear-num 95.7%

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

      2. frac-times 89.5%

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

      3. *-un-lft-identity 89.5%

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

      4. add-sqr-sqrt 40.0%

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

      5. sqrt-unprod 72.8%

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

      6. sqr-neg 72.8%

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

      7. sqrt-unprod 49.5%

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

      8. add-sqr-sqrt 84.3%

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

   10. Applied egg-rr 84.3%

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

   if -1.65000000000000008e181 < u < 6.8e202

   1. Initial program 70.1%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around inf 63.2%

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

      1. mul-1-neg 63.2%

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

      2. frac-2neg 63.2%

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

      3. distribute-neg-frac2 63.2%

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

      4. add-sqr-sqrt 29.5%

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

      5. sqrt-unprod 32.7%

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

      6. sqr-neg 32.7%

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

      7. sqrt-unprod 9.9%

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

      8. add-sqr-sqrt 16.5%

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

      9. distribute-neg-in 16.5%

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

      10. add-sqr-sqrt 8.1%

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

      11. sqrt-unprod 32.6%

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

      12. sqr-neg 32.6%

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

      13. sqrt-unprod 29.3%

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

      14. add-sqr-sqrt 64.4%

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

      15. sub-neg 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. sub-neg 64.4%

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

      2. distribute-neg-in 64.4%

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

      3. remove-double-neg 64.4%

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

   9. Simplified 64.4%

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

   10. Taylor expanded in v around 0 64.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+181} \lor \neg \left(u \leq 6.8 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e+198) (/ v (- u t1)) (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+198) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - 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 <= (-1.2d+198)) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+198) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e+198:
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e+198)
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e+198)
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e+198], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.2000000000000001e198

   1. Initial program 34.2%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. frac-2neg 96.4%

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

      3. distribute-neg-frac2 96.4%

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

      4. add-sqr-sqrt 38.4%

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

      5. sqrt-unprod 51.7%

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

      6. sqr-neg 51.7%

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

      7. sqrt-unprod 24.2%

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

      8. add-sqr-sqrt 35.7%

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

      9. distribute-neg-in 35.7%

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

      10. add-sqr-sqrt 35.7%

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

      11. sqrt-unprod 36.5%

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

      12. sqr-neg 36.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 96.6%

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

      15. sub-neg 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-neg-in 96.6%

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

      3. remove-double-neg 96.6%

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

   9. Simplified 96.6%

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

   10. Taylor expanded in v around 0 96.6%

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

   if -1.2000000000000001e198 < t1

   1. Initial program 76.9%

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

      1. associate-/l* 81.9%

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

   3. Simplified 81.9%

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

      1. associate-/r* 92.6%

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

      2. div-inv 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. associate-*r/ 92.6%

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

      2. *-rgt-identity 92.6%

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

   8. Simplified 92.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+181} \lor \neg \left(u \leq 8.8 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.4e+181) (not (<= u 8.8e+198))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.4e+181) || !(u <= 8.8e+198)) {
		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 <= (-2.4d+181)) .or. (.not. (u <= 8.8d+198))) 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 <= -2.4e+181) || !(u <= 8.8e+198)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.4e+181) or not (u <= 8.8e+198):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.4e+181) || !(u <= 8.8e+198))
		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 <= -2.4e+181) || ~((u <= 8.8e+198)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.4e+181], N[Not[LessEqual[u, 8.8e+198]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.40000000000000002e181 or 8.7999999999999998e198 < u

   1. Initial program 84.6%

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

      1. times-frac 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t1 around inf 45.2%

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

      1. mul-1-neg 45.2%

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

      2. frac-2neg 45.2%

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

      3. distribute-neg-frac2 45.2%

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

      4. add-sqr-sqrt 21.5%

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

      5. sqrt-unprod 44.8%

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

      6. sqr-neg 44.8%

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

      7. sqrt-unprod 23.7%

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

      8. add-sqr-sqrt 43.2%

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

      9. distribute-neg-in 43.2%

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

      10. add-sqr-sqrt 19.7%

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

      11. sqrt-unprod 43.2%

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

      12. sqr-neg 43.2%

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

      13. sqrt-unprod 23.5%

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

      14. add-sqr-sqrt 45.3%

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

      15. sub-neg 45.3%

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

   7. Applied egg-rr 45.3%

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

      1. sub-neg 45.3%

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

      2. distribute-neg-in 45.3%

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

      3. remove-double-neg 45.3%

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

   9. Simplified 45.3%

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

   10. Taylor expanded in t1 around 0 43.2%

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

   if -2.40000000000000002e181 < u < 8.7999999999999998e198

   1. Initial program 70.0%

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

      1. associate-*l/ 75.6%

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

      2. *-commutative 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in t1 around inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+181} \lor \neg \left(u \leq 8.8 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+181}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.9e+181) (/ v (- u)) (if (<= u 2.5e+200) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.9e+181) {
		tmp = v / -u;
	} else if (u <= 2.5e+200) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.9d+181)) then
        tmp = v / -u
    else if (u <= 2.5d+200) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.9e+181) {
		tmp = v / -u;
	} else if (u <= 2.5e+200) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.9e+181:
		tmp = v / -u
	elif u <= 2.5e+200:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.9e+181)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 2.5e+200)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.9e+181)
		tmp = v / -u;
	elseif (u <= 2.5e+200)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.9e+181], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 2.5e+200], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.9000000000000001e181

   1. Initial program 86.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 41.3%

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

   6. Taylor expanded in t1 around 0 38.2%

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

      1. associate-*r/ 38.2%

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

      2. mul-1-neg 38.2%

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

   8. Simplified 38.2%

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

   if -1.9000000000000001e181 < u < 2.50000000000000009e200

   1. Initial program 70.3%

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

      1. associate-*l/ 75.9%

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

      2. *-commutative 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in t1 around inf 62.9%

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

      1. associate-*r/ 62.9%

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

      2. neg-mul-1 62.9%

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

   7. Simplified 62.9%

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

   if 2.50000000000000009e200 < u

   1. Initial program 75.9%

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

      1. times-frac 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. +-commutative 98.1%

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

      5. distribute-neg-in 98.1%

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

      6. unsub-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around inf 49.4%

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

      1. mul-1-neg 49.4%

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

      2. frac-2neg 49.4%

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

      3. distribute-neg-frac2 49.4%

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

      4. add-sqr-sqrt 13.7%

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

      5. sqrt-unprod 48.9%

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

      6. sqr-neg 48.9%

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

      7. sqrt-unprod 35.7%

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

      8. add-sqr-sqrt 50.0%

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

      9. distribute-neg-in 50.0%

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

      10. add-sqr-sqrt 19.9%

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

      11. sqrt-unprod 49.7%

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

      12. sqr-neg 49.7%

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

      13. sqrt-unprod 30.1%

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

      14. add-sqr-sqrt 50.0%

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

      15. sub-neg 50.0%

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

   7. Applied egg-rr 50.0%

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

      1. sub-neg 50.0%

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

      2. distribute-neg-in 50.0%

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

      3. remove-double-neg 50.0%

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

   9. Simplified 50.0%

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

   10. Taylor expanded in t1 around 0 50.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+181}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 22.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+205} \lor \neg \left(t1 \leq 2.6 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.4e+205) (not (<= t1 2.6e+76))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.4e+205) || !(t1 <= 2.6e+76)) {
		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 <= (-7.4d+205)) .or. (.not. (t1 <= 2.6d+76))) 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 <= -7.4e+205) || !(t1 <= 2.6e+76)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.4e+205) or not (t1 <= 2.6e+76):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.4e+205) || !(t1 <= 2.6e+76))
		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 <= -7.4e+205) || ~((t1 <= 2.6e+76)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.4e+205], N[Not[LessEqual[t1, 2.6e+76]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.39999999999999961e205 or 2.5999999999999999e76 < t1

   1. Initial program 54.9%

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

      1. associate-*l/ 57.7%

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

      2. *-commutative 57.7%

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

   3. Simplified 57.7%

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

   5. Taylor expanded in t1 around inf 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

   7. Simplified 97.1%

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

      1. neg-sub0 97.1%

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

      2. sub-neg 97.1%

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

      3. add-sqr-sqrt 38.7%

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

      4. sqrt-unprod 54.6%

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

      5. sqr-neg 54.6%

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

      6. sqrt-unprod 26.7%

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

      7. add-sqr-sqrt 39.7%

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

   9. Applied egg-rr 39.7%

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

       1. +-lft-identity 39.7%

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

   11. Simplified 39.7%

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

   if -7.39999999999999961e205 < t1 < 2.5999999999999999e76

   1. Initial program 78.2%

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

      1. times-frac 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 inf 48.0%

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

      1. mul-1-neg 48.0%

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

      2. frac-2neg 48.0%

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

      3. distribute-neg-frac2 48.0%

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

      4. add-sqr-sqrt 24.9%

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

      5. sqrt-unprod 28.2%

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

      6. sqr-neg 28.2%

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

      7. sqrt-unprod 7.3%

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

      8. add-sqr-sqrt 14.7%

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

      9. distribute-neg-in 14.7%

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

      10. add-sqr-sqrt 8.2%

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

      11. sqrt-unprod 26.6%

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

      12. sqr-neg 26.6%

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

      13. sqrt-unprod 18.5%

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

      14. add-sqr-sqrt 49.4%

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

      15. sub-neg 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. sub-neg 49.4%

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

      2. distribute-neg-in 49.4%

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

      3. remove-double-neg 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in t1 around 0 15.4%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+205} \lor \neg \left(t1 \leq 2.6 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. times-frac 98.9%

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

   2. distribute-frac-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. distribute-neg-in 98.9%

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

   6. unsub-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in t1 around inf 59.9%

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

   1. mul-1-neg 59.9%

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

   2. frac-2neg 59.9%

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

   3. distribute-neg-frac2 59.9%

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

   4. add-sqr-sqrt 28.2%

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

   5. sqrt-unprod 34.5%

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

   6. sqr-neg 34.5%

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

   7. sqrt-unprod 12.0%

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

   8. add-sqr-sqrt 20.8%

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

   9. distribute-neg-in 20.8%

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

   10. add-sqr-sqrt 9.8%

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

   11. sqrt-unprod 34.2%

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

   12. sqr-neg 34.2%

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

   13. sqrt-unprod 28.4%

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

   14. add-sqr-sqrt 60.9%

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

   15. sub-neg 60.9%

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

7. Applied egg-rr 60.9%

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

   1. sub-neg 60.9%

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

   2. distribute-neg-in 60.9%

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

   3. remove-double-neg 60.9%

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

9. Simplified 60.9%

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

10. Taylor expanded in v around 0 60.9%

    \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
11. Add Preprocessing

Alternative 14: 14.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. associate-*l/ 77.2%

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

   2. *-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in t1 around inf 54.2%

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

   1. associate-*r/ 54.2%

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

   2. neg-mul-1 54.2%

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

7. Simplified 54.2%

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

   1. neg-sub0 54.2%

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

   2. sub-neg 54.2%

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

   3. add-sqr-sqrt 25.5%

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

   4. sqrt-unprod 31.2%

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

   5. sqr-neg 31.2%

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

   6. sqrt-unprod 7.7%

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

   7. add-sqr-sqrt 12.6%

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

9. Applied egg-rr 12.6%

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

    1. +-lft-identity 12.6%

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

11. Simplified 12.6%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))