Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.0%

Time: 10.7s

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 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 79.1%

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

   1. times-frac 96.6%

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

3. Simplified 96.6%

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

5. Final simplification 96.6%

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

Alternative 2: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{-92}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6e-92)
   (* (/ t1 u) (/ (- v) u))
   (if (<= u 8.2e-69) (- (/ v t1)) (/ t1 (* u (/ (- t1 u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e-92) {
		tmp = (t1 / u) * (-v / u);
	} else if (u <= 8.2e-69) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6d-92)) then
        tmp = (t1 / u) * (-v / u)
    else if (u <= 8.2d-69) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * ((t1 - u) / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e-92) {
		tmp = (t1 / u) * (-v / u);
	} else if (u <= 8.2e-69) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6e-92:
		tmp = (t1 / u) * (-v / u)
	elif u <= 8.2e-69:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * ((t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6e-92)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif (u <= 8.2e-69)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6e-92)
		tmp = (t1 / u) * (-v / u);
	elseif (u <= 8.2e-69)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * ((t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6e-92], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.2e-69], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.00000000000000027e-92

   1. Initial program 86.7%

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

      1. times-frac 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in t1 around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. mul-1-neg 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in t1 around 0 77.8%

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

   if -6.00000000000000027e-92 < u < 8.1999999999999998e-69

   1. Initial program 68.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. neg-mul-1 90.5%

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

   7. Simplified 90.5%

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

   if 8.1999999999999998e-69 < u

   1. Initial program 84.0%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. mul-1-neg 78.2%

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

   7. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. clear-num 78.2%

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

      3. frac-times 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. add-sqr-sqrt 35.4%

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

      6. sqrt-unprod 48.1%

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

      7. sqr-neg 48.1%

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

      8. sqrt-unprod 25.3%

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

      9. add-sqr-sqrt 48.0%

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

      10. frac-2neg 48.0%

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

      11. add-sqr-sqrt 22.3%

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

      12. sqrt-unprod 54.1%

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

      13. sqr-neg 54.1%

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

      14. sqrt-unprod 40.5%

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

      15. add-sqr-sqrt 79.6%

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

      16. distribute-neg-in 79.6%

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

      17. add-sqr-sqrt 35.5%

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

      18. sqrt-unprod 79.6%

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

      19. sqr-neg 79.6%

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

      20. sqrt-unprod 43.8%

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

      21. add-sqr-sqrt 79.6%

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

      22. sub-neg 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{-92}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{v}{1 - \frac{u}{t1}}}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-71}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.8e-92)
   (/ (/ v (- 1.0 (/ u t1))) u)
   (if (<= u 9.5e-71) (- (/ v t1)) (/ t1 (* u (/ (- t1 u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e-92) {
		tmp = (v / (1.0 - (u / t1))) / u;
	} else if (u <= 9.5e-71) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.8d-92)) then
        tmp = (v / (1.0d0 - (u / t1))) / u
    else if (u <= 9.5d-71) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * ((t1 - u) / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e-92) {
		tmp = (v / (1.0 - (u / t1))) / u;
	} else if (u <= 9.5e-71) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.8e-92:
		tmp = (v / (1.0 - (u / t1))) / u
	elif u <= 9.5e-71:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * ((t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.8e-92)
		tmp = Float64(Float64(v / Float64(1.0 - Float64(u / t1))) / u);
	elseif (u <= 9.5e-71)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.8e-92)
		tmp = (v / (1.0 - (u / t1))) / u;
	elseif (u <= 9.5e-71)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * ((t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.8e-92], N[(N[(v / N[(1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 9.5e-71], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.79999999999999969e-92

   1. Initial program 86.7%

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

      1. times-frac 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in t1 around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. mul-1-neg 78.6%

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

   7. Simplified 78.6%

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

      1. associate-*l/ 80.9%

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

   9. Applied egg-rr 77.8%

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

   if -5.79999999999999969e-92 < u < 9.4999999999999994e-71

   1. Initial program 68.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. neg-mul-1 90.5%

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

   7. Simplified 90.5%

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

   if 9.4999999999999994e-71 < u

   1. Initial program 84.0%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. mul-1-neg 78.2%

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

   7. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. clear-num 78.2%

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

      3. frac-times 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. add-sqr-sqrt 35.4%

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

      6. sqrt-unprod 48.1%

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

      7. sqr-neg 48.1%

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

      8. sqrt-unprod 25.3%

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

      9. add-sqr-sqrt 48.0%

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

      10. frac-2neg 48.0%

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

      11. add-sqr-sqrt 22.3%

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

      12. sqrt-unprod 54.1%

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

      13. sqr-neg 54.1%

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

      14. sqrt-unprod 40.5%

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

      15. add-sqr-sqrt 79.6%

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

      16. distribute-neg-in 79.6%

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

      17. add-sqr-sqrt 35.5%

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

      18. sqrt-unprod 79.6%

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

      19. sqr-neg 79.6%

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

      20. sqrt-unprod 43.8%

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

      21. add-sqr-sqrt 79.6%

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

      22. sub-neg 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{v}{1 - \frac{u}{t1}}}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-71}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{-70}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.5e-124)
   (* (/ v (+ t1 u)) (/ (- t1) u))
   (if (<= u 1.25e-70) (- (/ v t1)) (/ t1 (* u (/ (- t1 u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e-124) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= 1.25e-70) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.5d-124)) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else if (u <= 1.25d-70) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * ((t1 - u) / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e-124) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else if (u <= 1.25e-70) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.5e-124:
		tmp = (v / (t1 + u)) * (-t1 / u)
	elif u <= 1.25e-70:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * ((t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.5e-124)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	elseif (u <= 1.25e-70)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.5e-124)
		tmp = (v / (t1 + u)) * (-t1 / u);
	elseif (u <= 1.25e-70)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * ((t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.5e-124], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.25e-70], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.50000000000000016e-124

   1. Initial program 87.0%

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

      1. times-frac 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. mul-1-neg 78.0%

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

   7. Simplified 78.0%

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

   if -5.50000000000000016e-124 < u < 1.25e-70

   1. Initial program 68.2%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   7. Simplified 91.3%

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

   if 1.25e-70 < u

   1. Initial program 84.0%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. mul-1-neg 78.2%

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

   7. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. clear-num 78.2%

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

      3. frac-times 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. add-sqr-sqrt 35.4%

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

      6. sqrt-unprod 48.1%

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

      7. sqr-neg 48.1%

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

      8. sqrt-unprod 25.3%

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

      9. add-sqr-sqrt 48.0%

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

      10. frac-2neg 48.0%

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

      11. add-sqr-sqrt 22.3%

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

      12. sqrt-unprod 54.1%

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

      13. sqr-neg 54.1%

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

      14. sqrt-unprod 40.5%

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

      15. add-sqr-sqrt 79.6%

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

      16. distribute-neg-in 79.6%

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

      17. add-sqr-sqrt 35.5%

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

      18. sqrt-unprod 79.6%

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

      19. sqr-neg 79.6%

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

      20. sqrt-unprod 43.8%

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

      21. add-sqr-sqrt 79.6%

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

      22. sub-neg 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{-70}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.5e-124)
   (/ (* v (/ (- t1) u)) (+ t1 u))
   (if (<= u 8.5e-69) (- (/ v t1)) (/ t1 (* u (/ (- t1 u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e-124) {
		tmp = (v * (-t1 / u)) / (t1 + u);
	} else if (u <= 8.5e-69) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.5d-124)) then
        tmp = (v * (-t1 / u)) / (t1 + u)
    else if (u <= 8.5d-69) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * ((t1 - u) / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e-124) {
		tmp = (v * (-t1 / u)) / (t1 + u);
	} else if (u <= 8.5e-69) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.5e-124:
		tmp = (v * (-t1 / u)) / (t1 + u)
	elif u <= 8.5e-69:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * ((t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.5e-124)
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / Float64(t1 + u));
	elseif (u <= 8.5e-69)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.5e-124)
		tmp = (v * (-t1 / u)) / (t1 + u);
	elseif (u <= 8.5e-69)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * ((t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.5e-124], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.5e-69], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.50000000000000016e-124

   1. Initial program 87.0%

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

      1. times-frac 94.6%

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

   3. Simplified 94.6%

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

      1. associate-*r/ 95.7%

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

      2. clear-num 95.7%

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

      3. associate-*l/ 95.8%

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

      4. *-un-lft-identity 95.8%

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

      5. frac-2neg 95.8%

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

      6. distribute-neg-in 95.8%

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

      7. add-sqr-sqrt 46.7%

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

      8. sqrt-unprod 82.8%

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

      9. sqr-neg 82.8%

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

      10. sqrt-unprod 39.2%

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

      11. add-sqr-sqrt 77.8%

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

      12. sub-neg 77.8%

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

      13. remove-double-neg 77.8%

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

   6. Applied egg-rr 77.8%

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

   7. Taylor expanded in t1 around 0 79.0%

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

      1. mul-1-neg 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*r/ 78.1%

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

      4. *-commutative 78.1%

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

      5. distribute-rgt-neg-in 78.1%

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

   9. Simplified 78.1%

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

   if -5.50000000000000016e-124 < u < 8.50000000000000046e-69

   1. Initial program 68.2%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   7. Simplified 91.3%

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

   if 8.50000000000000046e-69 < u

   1. Initial program 84.0%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. mul-1-neg 78.2%

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

   7. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. clear-num 78.2%

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

      3. frac-times 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. add-sqr-sqrt 35.4%

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

      6. sqrt-unprod 48.1%

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

      7. sqr-neg 48.1%

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

      8. sqrt-unprod 25.3%

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

      9. add-sqr-sqrt 48.0%

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

      10. frac-2neg 48.0%

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

      11. add-sqr-sqrt 22.3%

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

      12. sqrt-unprod 54.1%

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

      13. sqr-neg 54.1%

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

      14. sqrt-unprod 40.5%

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

      15. add-sqr-sqrt 79.6%

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

      16. distribute-neg-in 79.6%

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

      17. add-sqr-sqrt 35.5%

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

      18. sqrt-unprod 79.6%

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

      19. sqr-neg 79.6%

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

      20. sqrt-unprod 43.8%

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

      21. add-sqr-sqrt 79.6%

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

      22. sub-neg 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{-92} \lor \neg \left(u \leq 1.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6e-92) (not (<= u 1.3e-68)))
   (* (/ t1 u) (/ (- v) u))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6e-92) || !(u <= 1.3e-68)) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6d-92)) .or. (.not. (u <= 1.3d-68))) then
        tmp = (t1 / u) * (-v / u)
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6e-92) || !(u <= 1.3e-68)) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6e-92) or not (u <= 1.3e-68):
		tmp = (t1 / u) * (-v / u)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6e-92) || !(u <= 1.3e-68))
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6e-92) || ~((u <= 1.3e-68)))
		tmp = (t1 / u) * (-v / u);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6e-92], N[Not[LessEqual[u, 1.3e-68]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -6.00000000000000027e-92 or 1.2999999999999999e-68 < u

   1. Initial program 85.5%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. mul-1-neg 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in t1 around 0 76.2%

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

   if -6.00000000000000027e-92 < u < 1.2999999999999999e-68

   1. Initial program 68.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. neg-mul-1 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{-92} \lor \neg \left(u \leq 1.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{-70}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.8e-92)
   (* (/ t1 u) (/ (- v) u))
   (if (<= u 4.7e-70) (- (/ v t1)) (/ t1 (* (- u) (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e-92) {
		tmp = (t1 / u) * (-v / u);
	} else if (u <= 4.7e-70) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (-u * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.8d-92)) then
        tmp = (t1 / u) * (-v / u)
    else if (u <= 4.7d-70) then
        tmp = -(v / t1)
    else
        tmp = t1 / (-u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e-92) {
		tmp = (t1 / u) * (-v / u);
	} else if (u <= 4.7e-70) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (-u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.8e-92:
		tmp = (t1 / u) * (-v / u)
	elif u <= 4.7e-70:
		tmp = -(v / t1)
	else:
		tmp = t1 / (-u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.8e-92)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif (u <= 4.7e-70)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(Float64(-u) * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.8e-92)
		tmp = (t1 / u) * (-v / u);
	elseif (u <= 4.7e-70)
		tmp = -(v / t1);
	else
		tmp = t1 / (-u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.8e-92], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.7e-70], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[((-u) * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.79999999999999969e-92

   1. Initial program 86.7%

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

      1. times-frac 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in t1 around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. mul-1-neg 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in t1 around 0 77.8%

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

   if -5.79999999999999969e-92 < u < 4.6999999999999998e-70

   1. Initial program 68.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. neg-mul-1 90.5%

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

   7. Simplified 90.5%

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

   if 4.6999999999999998e-70 < u

   1. Initial program 84.0%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. mul-1-neg 78.2%

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

   7. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. clear-num 78.2%

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

      3. frac-times 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. add-sqr-sqrt 35.4%

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

      6. sqrt-unprod 48.1%

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

      7. sqr-neg 48.1%

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

      8. sqrt-unprod 25.3%

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

      9. add-sqr-sqrt 48.0%

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

      10. frac-2neg 48.0%

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

      11. add-sqr-sqrt 22.3%

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

      12. sqrt-unprod 54.1%

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

      13. sqr-neg 54.1%

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

      14. sqrt-unprod 40.5%

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

      15. add-sqr-sqrt 79.6%

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

      16. distribute-neg-in 79.6%

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

      17. add-sqr-sqrt 35.5%

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

      18. sqrt-unprod 79.6%

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

      19. sqr-neg 79.6%

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

      20. sqrt-unprod 43.8%

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

      21. add-sqr-sqrt 79.6%

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

      22. sub-neg 79.6%

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

   9. Applied egg-rr 79.6%

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

   10. Taylor expanded in t1 around 0 78.3%

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

       1. neg-mul-1 78.3%

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

       2. distribute-neg-frac 78.3%

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

   12. Simplified 78.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{-70}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+165} \lor \neg \left(u \leq 1.9 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.2e+165) (not (<= u 1.9e+211)))
   (* (/ v u) -0.5)
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.2e+165) || !(u <= 1.9e+211)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6.2d+165)) .or. (.not. (u <= 1.9d+211))) then
        tmp = (v / u) * (-0.5d0)
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.2e+165) || !(u <= 1.9e+211)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.2e+165) or not (u <= 1.9e+211):
		tmp = (v / u) * -0.5
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.2e+165) || !(u <= 1.9e+211))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.2e+165) || ~((u <= 1.9e+211)))
		tmp = (v / u) * -0.5;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.2e+165], N[Not[LessEqual[u, 1.9e+211]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -6.2000000000000003e165 or 1.90000000000000008e211 < u

   1. Initial program 89.9%

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

      1. associate-/r* 95.8%

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

      2. *-commutative 95.8%

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

      3. associate-/l* 100.0%

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

      4. associate-/l/ 92.1%

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

      5. +-commutative 92.1%

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

      6. remove-double-neg 92.1%

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

      7. unsub-neg 92.1%

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

      8. div-sub 92.1%

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

      9. sub-neg 92.1%

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

      10. *-inverses 92.1%

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

      11. metadata-eval 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in t1 around inf 49.1%

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

      1. mul-1-neg 49.1%

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

      2. unsub-neg 49.1%

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

      3. *-commutative 49.1%

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

   7. Simplified 49.1%

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

   8. Taylor expanded in u around inf 49.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -6.2000000000000003e165 < u < 1.90000000000000008e211

   1. Initial program 76.6%

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

      1. times-frac 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in t1 around inf 60.0%

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

      1. associate-*r/ 60.0%

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

      2. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+165} \lor \neg \left(u \leq 1.9 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+173} \lor \neg \left(u \leq 2.1 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.9e+173) (not (<= u 2.1e+211))) (/ v u) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.9e+173) || !(u <= 2.1e+211)) {
		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 <= (-3.9d+173)) .or. (.not. (u <= 2.1d+211))) 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 <= -3.9e+173) || !(u <= 2.1e+211)) {
		tmp = v / u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.9e+173) or not (u <= 2.1e+211):
		tmp = v / u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.9e+173) || !(u <= 2.1e+211))
		tmp = Float64(v / u);
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.9e+173) || ~((u <= 2.1e+211)))
		tmp = v / u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.9e+173], N[Not[LessEqual[u, 2.1e+211]], $MachinePrecision]], N[(v / u), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -3.8999999999999998e173 or 2.1e211 < u

   1. Initial program 89.9%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. frac-times 92.1%

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

      4. *-un-lft-identity 92.1%

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

      5. add-sqr-sqrt 40.8%

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

      6. sqrt-unprod 79.2%

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

      7. sqr-neg 79.2%

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

      8. sqrt-unprod 49.2%

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

      9. add-sqr-sqrt 90.1%

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

      10. frac-2neg 90.1%

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

      11. add-sqr-sqrt 47.1%

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

      12. sqrt-unprod 75.2%

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

      13. sqr-neg 75.2%

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

      14. sqrt-unprod 43.0%

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

      15. add-sqr-sqrt 92.1%

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

      16. distribute-neg-in 92.1%

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

      17. add-sqr-sqrt 40.8%

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

      18. sqrt-unprod 92.1%

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

      19. sqr-neg 92.1%

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

      20. sqrt-unprod 51.3%

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

      21. add-sqr-sqrt 92.1%

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

      22. sub-neg 92.1%

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

   9. Applied egg-rr 92.1%

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

   10. Taylor expanded in t1 around inf 48.9%

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

   if -3.8999999999999998e173 < u < 2.1e211

   1. Initial program 76.6%

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

      1. times-frac 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in t1 around inf 60.0%

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

      1. associate-*r/ 60.0%

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

      2. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+173} \lor \neg \left(u \leq 2.1 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+167} \lor \neg \left(u \leq 1.85 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7e+167) (not (<= u 1.85e+211))) (/ (- v) u) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e+167) || !(u <= 1.85e+211)) {
		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 <= (-1.7d+167)) .or. (.not. (u <= 1.85d+211))) 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 <= -1.7e+167) || !(u <= 1.85e+211)) {
		tmp = -v / u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7e+167) or not (u <= 1.85e+211):
		tmp = -v / u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7e+167) || !(u <= 1.85e+211))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.7e+167) || ~((u <= 1.85e+211)))
		tmp = -v / u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7e+167], N[Not[LessEqual[u, 1.85e+211]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -1.7e167 or 1.85000000000000005e211 < u

   1. Initial program 89.9%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t1 around inf 49.1%

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

      1. associate-*r/ 49.1%

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

      2. neg-mul-1 49.1%

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

   10. Simplified 49.1%

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

   if -1.7e167 < u < 1.85000000000000005e211

   1. Initial program 76.6%

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

      1. times-frac 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in t1 around inf 60.0%

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

      1. associate-*r/ 60.0%

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

      2. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+167} \lor \neg \left(u \leq 1.85 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 20.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq 1.85 \cdot 10^{+130}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= t1 1.85e+130) (/ v u) (/ v t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= 1.85e+130) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= 1.85d+130) then
        tmp = v / u
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= 1.85e+130) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= 1.85e+130:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= 1.85e+130)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= 1.85e+130)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, 1.85e+130], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < 1.8500000000000001e130

   1. Initial program 83.3%

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

      1. times-frac 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t1 around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. mul-1-neg 60.7%

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

   7. Simplified 60.7%

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

      1. *-commutative 60.7%

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

      2. clear-num 60.6%

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

      3. frac-times 59.6%

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

      4. *-un-lft-identity 59.6%

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

      5. add-sqr-sqrt 31.3%

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

      6. sqrt-unprod 38.1%

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

      7. sqr-neg 38.1%

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

      8. sqrt-unprod 16.3%

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

      9. add-sqr-sqrt 35.0%

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

      10. frac-2neg 35.0%

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

      11. add-sqr-sqrt 17.0%

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

      12. sqrt-unprod 38.7%

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

      13. sqr-neg 38.7%

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

      14. sqrt-unprod 29.8%

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

      15. add-sqr-sqrt 59.6%

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

      16. distribute-neg-in 59.6%

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

      17. add-sqr-sqrt 31.4%

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

      18. sqrt-unprod 60.8%

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

      19. sqr-neg 60.8%

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

      20. sqrt-unprod 28.8%

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

      21. add-sqr-sqrt 58.9%

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

      22. sub-neg 58.9%

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

   9. Applied egg-rr 58.9%

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

   10. Taylor expanded in t1 around inf 16.0%

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

   if 1.8500000000000001e130 < t1

   1. Initial program 53.3%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 99.1%

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

      3. frac-2neg 99.1%

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

      4. frac-times 70.5%

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

      5. *-un-lft-identity 70.5%

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

      6. remove-double-neg 70.5%

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

      7. distribute-neg-in 70.5%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 43.5%

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

      10. sqr-neg 43.5%

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

      11. sqrt-unprod 43.2%

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

      12. add-sqr-sqrt 43.2%

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

      13. sub-neg 43.2%

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

   6. Applied egg-rr 43.2%

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

   7. Taylor expanded in t1 around inf 42.7%

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

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq 1.85 \cdot 10^{+130}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (* u -2.0) t1)))

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / ((u * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(u * -2.0) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

   1. associate-/r* 86.5%

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

   2. *-commutative 86.5%

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

   3. associate-/l* 97.4%

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

   4. associate-/l/ 93.9%

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

   5. +-commutative 93.9%

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

   6. remove-double-neg 93.9%

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

   7. unsub-neg 93.9%

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

   8. div-sub 93.9%

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

   9. sub-neg 93.9%

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

   10. *-inverses 93.9%

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

   11. metadata-eval 93.9%

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

3. Simplified 93.9%

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

5. Taylor expanded in t1 around inf 58.4%

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

   1. mul-1-neg 58.4%

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

   2. unsub-neg 58.4%

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

   3. *-commutative 58.4%

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

7. Simplified 58.4%

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

8. Final simplification 58.4%

   \[\leadsto \frac{v}{u \cdot -2 - t1} \]
9. Add Preprocessing

Alternative 13: 61.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. times-frac 96.6%

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

3. Simplified 96.6%

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

   1. associate-*r/ 97.3%

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

   2. clear-num 97.3%

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

   3. associate-*l/ 97.4%

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

   4. *-un-lft-identity 97.4%

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

   5. frac-2neg 97.4%

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

   6. distribute-neg-in 97.4%

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

   7. add-sqr-sqrt 46.0%

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

   8. sqrt-unprod 70.8%

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

   9. sqr-neg 70.8%

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

   10. sqrt-unprod 30.9%

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

   11. add-sqr-sqrt 57.9%

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

   12. sub-neg 57.9%

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

   13. remove-double-neg 57.9%

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

6. Applied egg-rr 57.9%

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

7. Taylor expanded in t1 around inf 19.6%

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

   1. div-inv 19.6%

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

   2. frac-2neg 19.6%

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

   3. metadata-eval 19.6%

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

   4. distribute-neg-in 19.6%

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

   5. add-sqr-sqrt 7.9%

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

   6. sqrt-unprod 33.9%

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

   7. sqr-neg 33.9%

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

   8. sqrt-unprod 31.5%

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

   9. add-sqr-sqrt 58.1%

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

   10. sub-neg 58.1%

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

9. Applied egg-rr 58.1%

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

    1. *-commutative 58.1%

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

    2. associate-*l/ 58.3%

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

    3. neg-mul-1 58.3%

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

11. Simplified 58.3%

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

12. Final simplification 58.3%

    \[\leadsto \frac{-v}{t1 - u} \]
13. 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 79.1%

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

   1. times-frac 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. clear-num 96.3%

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

   3. frac-2neg 96.3%

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

   4. frac-times 84.7%

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

   5. *-un-lft-identity 84.7%

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

   6. remove-double-neg 84.7%

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

   7. distribute-neg-in 84.7%

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

   8. add-sqr-sqrt 40.5%

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

   9. sqrt-unprod 67.3%

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

   10. sqr-neg 67.3%

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

   11. sqrt-unprod 29.7%

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

   12. add-sqr-sqrt 55.1%

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

   13. sub-neg 55.1%

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

6. Applied egg-rr 55.1%

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

7. Taylor expanded in t1 around inf 11.3%

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

8. Final simplification 11.3%

   \[\leadsto \frac{v}{t1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))