Rosa's DopplerBench

Percentage Accurate: 72.9% → 97.7%

Time: 9.4s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-v}{t1 + u}}{\frac{u}{t1} + 1} \end{array} \]

FPCore

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

C

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

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)) / ((u / t1) + 1.0d0)
end function

Java

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

Python

def code(u, v, t1):
	return (-v / (t1 + u)) / ((u / t1) + 1.0)

Julia

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

MATLAB

function tmp = code(u, v, t1)
	tmp = (-v / (t1 + u)) / ((u / t1) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. times-frac 98.1%

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

   2. neg-mul-1 98.1%

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

   3. associate-/l* 98.0%

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

   4. associate-*l/ 98.1%

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

   5. neg-mul-1 98.1%

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

   6. distribute-frac-neg 98.1%

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

   7. +-commutative 98.1%

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

   8. remove-double-neg 98.1%

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

   9. unsub-neg 98.1%

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

   10. div-sub 98.1%

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

   11. sub-neg 98.1%

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

   12. distribute-frac-neg 98.1%

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

   13. remove-double-neg 98.1%

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

   14. *-inverses 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 2: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-5} \lor \neg \left(u \leq 1.15 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7e-5) (not (<= u 1.15e+32)))
   (/ (* t1 (/ v (+ t1 u))) (- t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e-5) || !(u <= 1.15e+32)) {
		tmp = (t1 * (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 <= (-1.7d-5)) .or. (.not. (u <= 1.15d+32))) then
        tmp = (t1 * (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 <= -1.7e-5) || !(u <= 1.15e+32)) {
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7e-5) or not (u <= 1.15e+32):
		tmp = (t1 * (v / (t1 + u))) / (t1 - u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7e-5) || !(u <= 1.15e+32))
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(t1 - 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-5) || ~((u <= 1.15e+32)))
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7e-5], N[Not[LessEqual[u, 1.15e+32]], $MachinePrecision]], N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.7e-5 or 1.15e32 < u

   1. Initial program 76.5%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

      1. frac-2neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. add-sqr-sqrt 42.8%

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

      6. sqrt-unprod 85.0%

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

      7. sqr-neg 85.0%

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

      8. sqrt-unprod 48.4%

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

      9. add-sqr-sqrt 86.3%

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

      10. sub-neg 86.3%

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

   5. Applied egg-rr 86.3%

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

   if -1.7e-5 < u < 1.15e32

   1. Initial program 73.8%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 78.4%

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

      1. mul-1-neg 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-5} \lor \neg \left(u \leq 1.15 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+68)
   (/ (/ t1 (/ u v)) (- t1 u))
   (if (<= u 1.25e+56) (/ (- v) t1) (* (/ (- t1) (+ t1 u)) (/ v u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+68) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= 1.25e+56) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (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 (u <= (-3.8d+68)) then
        tmp = (t1 / (u / v)) / (t1 - u)
    else if (u <= 1.25d+56) then
        tmp = -v / t1
    else
        tmp = (-t1 / (t1 + u)) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+68) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= 1.25e+56) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e+68:
		tmp = (t1 / (u / v)) / (t1 - u)
	elif u <= 1.25e+56:
		tmp = -v / t1
	else:
		tmp = (-t1 / (t1 + u)) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+68)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(t1 - u));
	elseif (u <= 1.25e+56)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e+68)
		tmp = (t1 / (u / v)) / (t1 - u);
	elseif (u <= 1.25e+56)
		tmp = -v / t1;
	else
		tmp = (-t1 / (t1 + u)) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+68], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.25e+56], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.8000000000000001e68

   1. Initial program 66.4%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

      1. frac-2neg 98.1%

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

      2. remove-double-neg 98.1%

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

      3. associate-*l/ 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. add-sqr-sqrt 39.2%

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

      6. sqrt-unprod 85.0%

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

      7. sqr-neg 85.0%

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

      8. sqrt-unprod 55.1%

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

      9. add-sqr-sqrt 90.5%

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

      10. sub-neg 90.5%

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

   5. Applied egg-rr 90.5%

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

   6. Taylor expanded in t1 around 0 81.9%

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

      1. associate-/l* 90.7%

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

   8. Simplified 90.7%

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

   if -3.8000000000000001e68 < u < 1.25000000000000006e56

   1. Initial program 75.4%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 75.8%

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

      1. mul-1-neg 75.8%

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

   6. Simplified 75.8%

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

   if 1.25000000000000006e56 < u

   1. Initial program 82.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 85.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 4: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{v}{u}}{\frac{u}{t1} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.9e+68)
   (/ (/ t1 (/ u v)) (- t1 u))
   (if (<= u 1.25e+56) (/ (- v) t1) (/ (- (/ v u)) (+ (/ u t1) 1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.9e+68) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= 1.25e+56) {
		tmp = -v / t1;
	} else {
		tmp = -(v / u) / ((u / t1) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.9d+68)) then
        tmp = (t1 / (u / v)) / (t1 - u)
    else if (u <= 1.25d+56) then
        tmp = -v / t1
    else
        tmp = -(v / u) / ((u / t1) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.9e+68) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= 1.25e+56) {
		tmp = -v / t1;
	} else {
		tmp = -(v / u) / ((u / t1) + 1.0);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.9e+68:
		tmp = (t1 / (u / v)) / (t1 - u)
	elif u <= 1.25e+56:
		tmp = -v / t1
	else:
		tmp = -(v / u) / ((u / t1) + 1.0)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.9e+68)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(t1 - u));
	elseif (u <= 1.25e+56)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-Float64(v / u)) / Float64(Float64(u / t1) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.9e+68)
		tmp = (t1 / (u / v)) / (t1 - u);
	elseif (u <= 1.25e+56)
		tmp = -v / t1;
	else
		tmp = -(v / u) / ((u / t1) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.9e+68], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.25e+56], N[((-v) / t1), $MachinePrecision], N[((-N[(v / u), $MachinePrecision]) / N[(N[(u / t1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.90000000000000019e68

   1. Initial program 66.4%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

      1. frac-2neg 98.1%

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

      2. remove-double-neg 98.1%

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

      3. associate-*l/ 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. add-sqr-sqrt 39.2%

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

      6. sqrt-unprod 85.0%

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

      7. sqr-neg 85.0%

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

      8. sqrt-unprod 55.1%

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

      9. add-sqr-sqrt 90.5%

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

      10. sub-neg 90.5%

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

   5. Applied egg-rr 90.5%

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

   6. Taylor expanded in t1 around 0 81.9%

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

      1. associate-/l* 90.7%

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

   8. Simplified 90.7%

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

   if -3.90000000000000019e68 < u < 1.25000000000000006e56

   1. Initial program 75.4%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 75.8%

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

      1. mul-1-neg 75.8%

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

   6. Simplified 75.8%

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

   if 1.25000000000000006e56 < u

   1. Initial program 82.6%

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

      1. times-frac 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. associate-*l/ 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 85.2%

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

      1. associate-*r/ 85.2%

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

      2. neg-mul-1 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{v}{u}}{\frac{u}{t1} + 1}\\ \end{array} \]

Alternative 5: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.35 \cdot 10^{+86} \lor \neg \left(u \leq 1.25 \cdot 10^{+56}\right):\\ \;\;\;\;\left(-\frac{v}{u}\right) \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.35e+86) (not (<= u 1.25e+56)))
   (* (- (/ v u)) (/ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.35e+86) || !(u <= 1.25e+56)) {
		tmp = -(v / 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 <= (-2.35d+86)) .or. (.not. (u <= 1.25d+56))) then
        tmp = -(v / 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 <= -2.35e+86) || !(u <= 1.25e+56)) {
		tmp = -(v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.35e+86) or not (u <= 1.25e+56):
		tmp = -(v / u) * (t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.35e+86) || !(u <= 1.25e+56))
		tmp = Float64(Float64(-Float64(v / u)) * Float64(t1 / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.35e+86) || ~((u <= 1.25e+56)))
		tmp = -(v / u) * (t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.35e+86], N[Not[LessEqual[u, 1.25e+56]], $MachinePrecision]], N[((-N[(v / u), $MachinePrecision]) * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.3500000000000001e86 or 1.25000000000000006e56 < u

   1. Initial program 73.5%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around 0 87.4%

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

   5. Taylor expanded in t1 around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. distribute-neg-frac 87.2%

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

   7. Simplified 87.2%

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

   if -2.3500000000000001e86 < u < 1.25000000000000006e56

   1. Initial program 76.0%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 75.8%

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

      1. mul-1-neg 75.8%

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

   6. Simplified 75.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.35 \cdot 10^{+86} \lor \neg \left(u \leq 1.25 \cdot 10^{+56}\right):\\ \;\;\;\;\left(-\frac{v}{u}\right) \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.8 \cdot 10^{-5} \lor \neg \left(u \leq 1.25 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.8e-5) (not (<= u 1.25e+56)))
   (/ (/ (- t1) u) (/ u v))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.8e-5) || !(u <= 1.25e+56)) {
		tmp = (-t1 / u) / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.8d-5)) .or. (.not. (u <= 1.25d+56))) then
        tmp = (-t1 / u) / (u / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.8e-5) || !(u <= 1.25e+56)) {
		tmp = (-t1 / u) / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.8e-5) or not (u <= 1.25e+56):
		tmp = (-t1 / u) / (u / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.8e-5) || !(u <= 1.25e+56))
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.8e-5) || ~((u <= 1.25e+56)))
		tmp = (-t1 / u) / (u / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.8e-5], N[Not[LessEqual[u, 1.25e+56]], $MachinePrecision]], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.80000000000000005e-5 or 1.25000000000000006e56 < u

   1. Initial program 76.3%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around 0 82.8%

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

      1. frac-2neg 82.8%

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

      2. clear-num 83.2%

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

      3. frac-times 80.6%

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

      4. remove-double-neg 80.6%

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

      5. *-commutative 80.6%

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

      6. *-un-lft-identity 80.6%

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

      7. distribute-neg-in 80.6%

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

      8. add-sqr-sqrt 33.9%

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

      9. sqrt-unprod 78.8%

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

      10. sqr-neg 78.8%

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

      11. sqrt-unprod 46.5%

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

      12. add-sqr-sqrt 80.2%

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

      13. sub-neg 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. associate-/r* 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in t1 around 0 83.2%

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

       1. mul-1-neg 83.2%

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

       2. distribute-neg-frac 83.2%

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

   11. Simplified 83.2%

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

   if -1.80000000000000005e-5 < u < 1.25000000000000006e56

   1. Initial program 74.0%

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

      1. times-frac 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t1 around inf 77.8%

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

      1. mul-1-neg 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.8 \cdot 10^{-5} \lor \neg \left(u \leq 1.25 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+68)
   (/ (/ t1 (/ u v)) (- t1 u))
   (if (<= u 3.5e+56) (/ (- v) t1) (/ (/ (- t1) u) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+68) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= 3.5e+56) {
		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 <= (-3.8d+68)) then
        tmp = (t1 / (u / v)) / (t1 - u)
    else if (u <= 3.5d+56) 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 <= -3.8e+68) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= 3.5e+56) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e+68:
		tmp = (t1 / (u / v)) / (t1 - u)
	elif u <= 3.5e+56:
		tmp = -v / t1
	else:
		tmp = (-t1 / u) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+68)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(t1 - u));
	elseif (u <= 3.5e+56)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e+68)
		tmp = (t1 / (u / v)) / (t1 - u);
	elseif (u <= 3.5e+56)
		tmp = -v / t1;
	else
		tmp = (-t1 / u) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+68], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.5e+56], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.8000000000000001e68

   1. Initial program 66.4%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

      1. frac-2neg 98.1%

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

      2. remove-double-neg 98.1%

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

      3. associate-*l/ 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. add-sqr-sqrt 39.2%

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

      6. sqrt-unprod 85.0%

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

      7. sqr-neg 85.0%

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

      8. sqrt-unprod 55.1%

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

      9. add-sqr-sqrt 90.5%

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

      10. sub-neg 90.5%

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

   5. Applied egg-rr 90.5%

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

   6. Taylor expanded in t1 around 0 81.9%

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

      1. associate-/l* 90.7%

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

   8. Simplified 90.7%

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

   if -3.8000000000000001e68 < u < 3.49999999999999999e56

   1. Initial program 75.4%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 75.8%

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

      1. mul-1-neg 75.8%

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

   6. Simplified 75.8%

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

   if 3.49999999999999999e56 < u

   1. Initial program 82.6%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 85.1%

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

      1. frac-2neg 85.1%

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

      2. clear-num 85.1%

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

      3. frac-times 82.5%

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

      4. remove-double-neg 82.5%

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

      5. *-commutative 82.5%

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

      6. *-un-lft-identity 82.5%

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

      7. distribute-neg-in 82.5%

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

      8. add-sqr-sqrt 35.7%

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

      9. sqrt-unprod 80.3%

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

      10. sqr-neg 80.3%

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

      11. sqrt-unprod 46.2%

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

      12. add-sqr-sqrt 81.9%

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

      13. sub-neg 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-/r* 84.6%

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

   8. Simplified 84.6%

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

   9. Taylor expanded in t1 around 0 84.9%

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

       1. mul-1-neg 84.9%

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

       2. distribute-neg-frac 84.9%

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

   11. Simplified 84.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \]

Alternative 8: 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 75.0%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 9: 68.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.3 \cdot 10^{+87} \lor \neg \left(u \leq 8.5 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.3e+87) (not (<= u 8.5e+89)))
   (/ t1 (/ u (/ v u)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.3e+87) || !(u <= 8.5e+89)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.3d+87)) .or. (.not. (u <= 8.5d+89))) then
        tmp = t1 / (u / (v / u))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.3e+87) || !(u <= 8.5e+89)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.3e+87) or not (u <= 8.5e+89):
		tmp = t1 / (u / (v / u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.3e+87) || !(u <= 8.5e+89))
		tmp = Float64(t1 / Float64(u / Float64(v / u)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.3e+87) || ~((u <= 8.5e+89)))
		tmp = t1 / (u / (v / u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.3e+87], N[Not[LessEqual[u, 8.5e+89]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.3000000000000001e87 or 8.50000000000000045e89 < u

   1. Initial program 71.2%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around 0 89.3%

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

   5. Taylor expanded in t1 around 0 89.2%

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

      1. mul-1-neg 89.2%

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

      2. distribute-neg-frac 89.2%

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

   7. Simplified 89.2%

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

      1. associate-*l/ 90.3%

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

      2. associate-/l* 84.1%

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

      3. add-sqr-sqrt 31.5%

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

      4. sqrt-unprod 61.0%

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

      5. sqr-neg 61.0%

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

      6. sqrt-unprod 41.1%

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

      7. add-sqr-sqrt 62.9%

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

   9. Applied egg-rr 62.9%

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

   if -3.3000000000000001e87 < u < 8.50000000000000045e89

   1. Initial program 77.1%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around inf 75.1%

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

   5. Taylor expanded in v around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

      3. +-commutative 74.5%

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

   7. Simplified 74.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.3 \cdot 10^{+87} \lor \neg \left(u \leq 8.5 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 10: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 8.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.2e+145)
   (/ v u)
   (if (<= u 8.6e+131) (/ (- v) t1) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.2e+145) {
		tmp = v / u;
	} else if (u <= 8.6e+131) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.2d+145)) then
        tmp = v / u
    else if (u <= 8.6d+131) then
        tmp = -v / t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.2e+145) {
		tmp = v / u;
	} else if (u <= 8.6e+131) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.2e+145:
		tmp = v / u
	elif u <= 8.6e+131:
		tmp = -v / t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.2e+145)
		tmp = Float64(v / u);
	elseif (u <= 8.6e+131)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.2e+145)
		tmp = v / u;
	elseif (u <= 8.6e+131)
		tmp = -v / t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.2e+145], N[(v / u), $MachinePrecision], If[LessEqual[u, 8.6e+131], N[((-v) / t1), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.20000000000000008e145

   1. Initial program 60.0%

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

      1. times-frac 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t1 around inf 51.8%

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

      1. frac-2neg 51.8%

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

      2. frac-2neg 51.8%

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

      3. frac-times 41.9%

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

      4. remove-double-neg 41.9%

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

      5. distribute-neg-in 41.9%

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

      6. add-sqr-sqrt 8.9%

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

      7. sqrt-unprod 41.9%

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

      8. sqr-neg 41.9%

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

      9. sqrt-unprod 33.0%

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

      10. add-sqr-sqrt 41.9%

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

      11. sub-neg 41.9%

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

      12. add-sqr-sqrt 8.9%

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

      13. sqrt-unprod 39.3%

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

      14. sqr-neg 39.3%

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

      15. sqrt-unprod 33.2%

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

      16. add-sqr-sqrt 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. distribute-rgt-neg-out 41.8%

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

      2. *-commutative 41.8%

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

      3. distribute-frac-neg 41.8%

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

      4. times-frac 47.3%

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

      5. *-inverses 47.3%

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

      6. associate-*r/ 47.3%

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

      7. metadata-eval 47.3%

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

      8. associate-*r* 47.3%

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

      9. *-commutative 47.3%

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

      10. associate-*r* 47.3%

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

      11. *-commutative 47.3%

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

      12. associate-*l* 47.3%

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

      13. metadata-eval 47.3%

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

      14. *-rgt-identity 47.3%

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

      15. distribute-frac-neg 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in t1 around 0 42.5%

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

   if -3.20000000000000008e145 < u < 8.6000000000000003e131

   1. Initial program 77.5%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 70.6%

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

      1. mul-1-neg 70.6%

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

   6. Simplified 70.6%

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

   if 8.6000000000000003e131 < u

   1. Initial program 78.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 58.2%

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

      1. expm1-log1p-u 57.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\right)\right)} \]

      2. expm1-udef 66.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\right)} - 1} \]

      3. frac-times 69.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot t1}}\right)} - 1 \]

      4. associate-/l* 70.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot t1}{v}}}\right)} - 1 \]

      5. add-sqr-sqrt 28.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{\frac{\left(t1 + u\right) \cdot t1}{v}}\right)} - 1 \]

      6. sqrt-unprod 63.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{\frac{\left(t1 + u\right) \cdot t1}{v}}\right)} - 1 \]

      7. sqr-neg 63.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{\frac{\left(t1 + u\right) \cdot t1}{v}}\right)} - 1 \]

      8. sqrt-unprod 41.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{\frac{\left(t1 + u\right) \cdot t1}{v}}\right)} - 1 \]

      9. add-sqr-sqrt 70.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{t1}}{\frac{\left(t1 + u\right) \cdot t1}{v}}\right)} - 1 \]

      10. *-commutative 70.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{t1}{\frac{\color{blue}{t1 \cdot \left(t1 + u\right)}}{v}}\right)} - 1 \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t1}{\frac{t1 \cdot \left(t1 + u\right)}{v}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 52.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t1}{\frac{t1 \cdot \left(t1 + u\right)}{v}}\right)\right)} \]

      2. expm1-log1p 52.4%

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

      3. associate-/l* 51.7%

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

      4. *-commutative 51.7%

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

      5. *-commutative 51.7%

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

      6. times-frac 49.6%

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

      7. *-inverses 49.6%

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

      8. *-rgt-identity 49.6%

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

      9. +-commutative 49.6%

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

   8. Simplified 49.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 8.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 11: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+145} \lor \neg \left(u \leq 1.4 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.9e+145) (not (<= u 1.4e+162))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e+145) || !(u <= 1.4e+162)) {
		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 <= (-2.9d+145)) .or. (.not. (u <= 1.4d+162))) 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 <= -2.9e+145) || !(u <= 1.4e+162)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.9e+145) or not (u <= 1.4e+162):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.9e+145) || !(u <= 1.4e+162))
		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 <= -2.9e+145) || ~((u <= 1.4e+162)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.9e+145], N[Not[LessEqual[u, 1.4e+162]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.9000000000000001e145 or 1.39999999999999995e162 < u

   1. Initial program 66.6%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in t1 around inf 56.3%

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

      1. frac-2neg 56.3%

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

      2. frac-2neg 56.3%

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

      3. frac-times 47.4%

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

      4. remove-double-neg 47.4%

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

      5. distribute-neg-in 47.4%

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

      6. add-sqr-sqrt 14.5%

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

      7. sqrt-unprod 47.4%

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

      8. sqr-neg 47.4%

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

      9. sqrt-unprod 33.0%

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

      10. add-sqr-sqrt 47.4%

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

      11. sub-neg 47.4%

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

      12. add-sqr-sqrt 14.5%

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

      13. sqrt-unprod 36.7%

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

      14. sqr-neg 36.7%

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

      15. sqrt-unprod 33.1%

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

      16. add-sqr-sqrt 47.4%

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

   6. Applied egg-rr 47.4%

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

      1. distribute-rgt-neg-out 47.4%

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

      2. *-commutative 47.4%

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

      3. distribute-frac-neg 47.4%

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

      4. times-frac 51.0%

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

      5. *-inverses 51.0%

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

      6. associate-*r/ 51.0%

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

      7. metadata-eval 51.0%

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

      8. associate-*r* 51.0%

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

      9. *-commutative 51.0%

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

      10. associate-*r* 51.0%

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

      11. *-commutative 51.0%

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

      12. associate-*l* 51.0%

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

      13. metadata-eval 51.0%

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

      14. *-rgt-identity 51.0%

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

      15. distribute-frac-neg 51.0%

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

   8. Simplified 51.0%

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

   9. Taylor expanded in t1 around 0 46.7%

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

   if -2.9000000000000001e145 < u < 1.39999999999999995e162

   1. Initial program 78.0%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 69.7%

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

      1. mul-1-neg 69.7%

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

   6. Simplified 69.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+145} \lor \neg \left(u \leq 1.4 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+159}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.4e+145) (/ v u) (if (<= u 9e+159) (/ (- v) t1) (- (/ v u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.4e+145) {
		tmp = v / u;
	} else if (u <= 9e+159) {
		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 <= (-6.4d+145)) then
        tmp = v / u
    else if (u <= 9d+159) 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 <= -6.4e+145) {
		tmp = v / u;
	} else if (u <= 9e+159) {
		tmp = -v / t1;
	} else {
		tmp = -(v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.4e+145:
		tmp = v / u
	elif u <= 9e+159:
		tmp = -v / t1
	else:
		tmp = -(v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.4e+145)
		tmp = Float64(v / u);
	elseif (u <= 9e+159)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(-Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.4e+145)
		tmp = v / u;
	elseif (u <= 9e+159)
		tmp = -v / t1;
	else
		tmp = -(v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.4e+145], N[(v / u), $MachinePrecision], If[LessEqual[u, 9e+159], N[((-v) / t1), $MachinePrecision], (-N[(v / u), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if u < -6.40000000000000015e145

   1. Initial program 60.0%

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

      1. times-frac 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t1 around inf 51.8%

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

      1. frac-2neg 51.8%

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

      2. frac-2neg 51.8%

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

      3. frac-times 41.9%

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

      4. remove-double-neg 41.9%

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

      5. distribute-neg-in 41.9%

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

      6. add-sqr-sqrt 8.9%

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

      7. sqrt-unprod 41.9%

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

      8. sqr-neg 41.9%

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

      9. sqrt-unprod 33.0%

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

      10. add-sqr-sqrt 41.9%

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

      11. sub-neg 41.9%

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

      12. add-sqr-sqrt 8.9%

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

      13. sqrt-unprod 39.3%

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

      14. sqr-neg 39.3%

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

      15. sqrt-unprod 33.2%

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

      16. add-sqr-sqrt 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. distribute-rgt-neg-out 41.8%

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

      2. *-commutative 41.8%

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

      3. distribute-frac-neg 41.8%

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

      4. times-frac 47.3%

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

      5. *-inverses 47.3%

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

      6. associate-*r/ 47.3%

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

      7. metadata-eval 47.3%

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

      8. associate-*r* 47.3%

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

      9. *-commutative 47.3%

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

      10. associate-*r* 47.3%

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

      11. *-commutative 47.3%

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

      12. associate-*l* 47.3%

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

      13. metadata-eval 47.3%

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

      14. *-rgt-identity 47.3%

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

      15. distribute-frac-neg 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in t1 around 0 42.5%

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

   if -6.40000000000000015e145 < u < 9.00000000000000053e159

   1. Initial program 78.0%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 69.7%

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

      1. mul-1-neg 69.7%

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

   6. Simplified 69.7%

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

   if 9.00000000000000053e159 < u

   1. Initial program 75.5%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 62.3%

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

   5. Taylor expanded in t1 around 0 52.5%

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

      1. associate-*r/ 52.5%

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

      2. neg-mul-1 52.5%

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

   7. Simplified 52.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+159}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \]

Alternative 13: 23.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+23} \lor \neg \left(t1 \leq 3.5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.2e+23) (not (<= t1 3.5e+69))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e+23) || !(t1 <= 3.5e+69)) {
		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 <= (-9.2d+23)) .or. (.not. (t1 <= 3.5d+69))) 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 <= -9.2e+23) || !(t1 <= 3.5e+69)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.2e+23) or not (t1 <= 3.5e+69):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.2e+23) || !(t1 <= 3.5e+69))
		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 <= -9.2e+23) || ~((t1 <= 3.5e+69)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.2e+23], N[Not[LessEqual[t1, 3.5e+69]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.2000000000000002e23 or 3.49999999999999987e69 < t1

   1. Initial program 59.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 83.3%

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

   5. Taylor expanded in u around inf 34.7%

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

   if -9.2000000000000002e23 < t1 < 3.49999999999999987e69

   1. Initial program 86.1%

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

      1. times-frac 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in t1 around inf 55.9%

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

      1. frac-2neg 55.9%

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

      2. frac-2neg 55.9%

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

      3. frac-times 54.7%

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

      4. remove-double-neg 54.7%

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

      5. distribute-neg-in 54.7%

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

      6. add-sqr-sqrt 24.1%

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

      7. sqrt-unprod 40.5%

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

      8. sqr-neg 40.5%

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

      9. sqrt-unprod 16.3%

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

      10. add-sqr-sqrt 23.2%

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

      11. sub-neg 23.2%

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

      12. add-sqr-sqrt 6.8%

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

      13. sqrt-unprod 25.8%

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

      14. sqr-neg 25.8%

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

      15. sqrt-unprod 29.1%

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

      16. add-sqr-sqrt 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. distribute-rgt-neg-out 53.4%

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

      2. *-commutative 53.4%

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

      3. distribute-frac-neg 53.4%

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

      4. times-frac 51.0%

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

      5. *-inverses 51.0%

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

      6. associate-*r/ 51.0%

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

      7. metadata-eval 51.0%

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

      8. associate-*r* 51.0%

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

      9. *-commutative 51.0%

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

      10. associate-*r* 51.0%

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

      11. *-commutative 51.0%

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

      12. associate-*l* 51.0%

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

      13. metadata-eval 51.0%

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

      14. *-rgt-identity 51.0%

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

      15. distribute-frac-neg 51.0%

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

   8. Simplified 51.0%

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

   9. Taylor expanded in t1 around 0 20.6%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+23} \lor \neg \left(t1 \leq 3.5 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14: 62.0% 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 75.0%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in t1 around inf 67.4%

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

5. Taylor expanded in v around 0 65.3%

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

   1. associate-*r/ 65.3%

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

   2. neg-mul-1 65.3%

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

   3. +-commutative 65.3%

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

7. Simplified 65.3%

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

8. Final simplification 65.3%

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

Alternative 15: 61.7% 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 75.0%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in t1 around inf 67.4%

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

   1. frac-2neg 67.4%

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

   2. frac-2neg 67.4%

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

   3. frac-times 54.3%

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

   4. remove-double-neg 54.3%

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

   5. distribute-neg-in 54.3%

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

   6. add-sqr-sqrt 25.5%

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

   7. sqrt-unprod 42.4%

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

   8. sqr-neg 42.4%

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

   9. sqrt-unprod 16.7%

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

   10. add-sqr-sqrt 28.3%

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

   11. sub-neg 28.3%

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

   12. add-sqr-sqrt 11.6%

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

   13. sqrt-unprod 33.5%

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

   14. sqr-neg 33.5%

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

   15. sqrt-unprod 27.8%

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

   16. add-sqr-sqrt 53.5%

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

6. Applied egg-rr 53.5%

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

   1. distribute-rgt-neg-out 53.5%

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

   2. *-commutative 53.5%

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

   3. distribute-frac-neg 53.5%

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

   4. times-frac 64.4%

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

   5. *-inverses 64.4%

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

   6. associate-*r/ 64.4%

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

   7. metadata-eval 64.4%

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

   8. associate-*r* 64.4%

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

   9. *-commutative 64.4%

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

   10. associate-*r* 64.4%

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

   11. *-commutative 64.4%

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

   12. associate-*l* 64.4%

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

   13. metadata-eval 64.4%

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

   14. *-rgt-identity 64.4%

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

   15. distribute-frac-neg 64.4%

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

8. Simplified 64.4%

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

   1. frac-2neg 64.4%

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

   2. div-inv 64.3%

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

   3. remove-double-neg 64.3%

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

10. Applied egg-rr 64.3%

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

    1. associate-*r/ 64.4%

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

    2. *-rgt-identity 64.4%

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

    3. neg-sub0 64.4%

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

    4. associate--r- 64.4%

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

    5. neg-sub0 64.4%

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

    6. +-commutative 64.4%

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

    7. sub-neg 64.4%

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

12. Simplified 64.4%

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

13. Final simplification 64.4%

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

Alternative 16: 14.4% 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 75.0%

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

   1. times-frac 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in t1 around inf 59.2%

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

5. Taylor expanded in u around inf 15.8%

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

6. Final simplification 15.8%

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

Reproduce

herbie shell --seed 2023299 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))