Rosa's DopplerBench

Percentage Accurate: 73.1% → 97.9%

Time: 8.0s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 73.4%

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

   1. times-frac 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 2: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+48} \lor \neg \left(u \leq 2.4 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.4e+48) (not (<= u 2.4e+91)))
   (/ (- t1) (* u (/ u v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.4e+48) || !(u <= 2.4e+91)) {
		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 <= (-2.4d+48)) .or. (.not. (u <= 2.4d+91))) 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 <= -2.4e+48) || !(u <= 2.4e+91)) {
		tmp = -t1 / (u * (u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.4e+48) or not (u <= 2.4e+91):
		tmp = -t1 / (u * (u / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.4e+48) || !(u <= 2.4e+91))
		tmp = Float64(Float64(-t1) / Float64(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 <= -2.4e+48) || ~((u <= 2.4e+91)))
		tmp = -t1 / (u * (u / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.4e+48], N[Not[LessEqual[u, 2.4e+91]], $MachinePrecision]], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.4000000000000001e48 or 2.39999999999999983e91 < u

   1. Initial program 79.3%

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

      1. times-frac 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around 0 74.6%

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

      1. associate-*r/ 74.6%

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

      2. neg-mul-1 74.6%

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

      3. unpow2 74.6%

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

   6. Simplified 74.6%

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

      1. distribute-lft-neg-in 74.6%

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

      2. times-frac 87.2%

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

      3. add-sqr-sqrt 38.7%

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

      4. sqrt-unprod 62.9%

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

      5. sqr-neg 62.9%

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

      6. sqrt-unprod 36.5%

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

      7. add-sqr-sqrt 64.3%

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

   8. Applied egg-rr 64.3%

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

      1. *-commutative 64.3%

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

   10. Simplified 64.3%

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

       1. frac-times 62.5%

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

   12. Applied egg-rr 62.5%

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

       1. frac-2neg 62.5%

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

       2. distribute-frac-neg 62.5%

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

       3. remove-double-neg 62.5%

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

       4. distribute-lft-neg-in 62.5%

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

       5. frac-times 64.3%

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

       6. frac-2neg 64.3%

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

       7. *-commutative 64.3%

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

       8. clear-num 65.2%

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

       9. frac-times 65.4%

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

       10. *-commutative 65.4%

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

       11. *-un-lft-identity 65.4%

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

       12. add-sqr-sqrt 29.8%

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

       13. sqrt-unprod 59.6%

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

       14. sqr-neg 59.6%

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

       15. sqrt-unprod 45.6%

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

       16. add-sqr-sqrt 89.0%

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

   14. Applied egg-rr 89.0%

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

   if -2.4000000000000001e48 < u < 2.39999999999999983e91

   1. Initial program 69.7%

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

      1. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t1 around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+48} \lor \neg \left(u \leq 2.4 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+47}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.25e+47)
   (/ (- t1) (* u (/ u v)))
   (if (<= u 2.4e+91) (/ (- v) t1) (/ (* t1 (/ v u)) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+47) {
		tmp = -t1 / (u * (u / v));
	} else if (u <= 2.4e+91) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.25d+47)) then
        tmp = -t1 / (u * (u / v))
    else if (u <= 2.4d+91) then
        tmp = -v / t1
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+47) {
		tmp = -t1 / (u * (u / v));
	} else if (u <= 2.4e+91) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.25e+47:
		tmp = -t1 / (u * (u / v))
	elif u <= 2.4e+91:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.25e+47)
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	elseif (u <= 2.4e+91)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.25e+47)
		tmp = -t1 / (u * (u / v));
	elseif (u <= 2.4e+91)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.25e+47], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.4e+91], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.25000000000000005e47

   1. Initial program 77.2%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t1 around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

      3. unpow2 71.9%

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

   6. Simplified 71.9%

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

      1. distribute-lft-neg-in 71.9%

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

      2. times-frac 90.1%

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

      3. add-sqr-sqrt 48.2%

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

      4. sqrt-unprod 62.5%

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

      5. sqr-neg 62.5%

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

      6. sqrt-unprod 29.5%

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

      7. add-sqr-sqrt 62.8%

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

   8. Applied egg-rr 62.8%

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

      1. *-commutative 62.8%

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

   10. Simplified 62.8%

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

       1. frac-times 59.4%

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

   12. Applied egg-rr 59.4%

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

       1. frac-2neg 59.4%

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

       2. distribute-frac-neg 59.4%

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

       3. remove-double-neg 59.4%

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

       4. distribute-lft-neg-in 59.4%

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

       5. frac-times 62.8%

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

       6. frac-2neg 62.8%

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

       7. *-commutative 62.8%

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

       8. clear-num 64.5%

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

       9. frac-times 64.6%

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

       10. *-commutative 64.6%

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

       11. *-un-lft-identity 64.6%

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

       12. add-sqr-sqrt 25.3%

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

       13. sqrt-unprod 60.2%

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

       14. sqr-neg 60.2%

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

       15. sqrt-unprod 48.0%

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

       16. add-sqr-sqrt 93.3%

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

   14. Applied egg-rr 93.3%

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

   if -1.25000000000000005e47 < u < 2.39999999999999983e91

   1. Initial program 69.7%

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

      1. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t1 around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

   if 2.39999999999999983e91 < u

   1. Initial program 81.7%

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

      1. times-frac 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in t1 around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. neg-mul-1 77.7%

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

      3. unpow2 77.7%

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

   6. Simplified 77.7%

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

      1. distribute-rgt-neg-in 77.7%

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

      2. times-frac 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. *-commutative 84.1%

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

      2. frac-2neg 84.1%

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

      3. remove-double-neg 84.1%

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

      4. associate-*l/ 82.0%

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

      5. add-sqr-sqrt 39.1%

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

      6. sqrt-unprod 56.8%

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

      7. sqr-neg 56.8%

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

      8. sqrt-unprod 32.8%

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

      9. add-sqr-sqrt 64.1%

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

      10. *-commutative 64.1%

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

      11. associate-*l/ 63.9%

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

      12. associate-*r/ 66.0%

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

      13. div-inv 66.0%

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

      14. add-sqr-sqrt 34.8%

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

      15. sqrt-unprod 59.0%

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

      16. sqr-neg 59.0%

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

      17. sqrt-unprod 42.9%

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

      18. add-sqr-sqrt 88.0%

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

      19. div-inv 88.0%

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+47}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+47}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 4.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+47)
   (/ (- t1) (/ (+ t1 u) (/ v u)))
   (if (<= u 4.9e+91) (/ (- v) t1) (/ (* t1 (/ v u)) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+47) {
		tmp = -t1 / ((t1 + u) / (v / u));
	} else if (u <= 4.9e+91) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.15d+47)) then
        tmp = -t1 / ((t1 + u) / (v / u))
    else if (u <= 4.9d+91) then
        tmp = -v / t1
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+47) {
		tmp = -t1 / ((t1 + u) / (v / u));
	} else if (u <= 4.9e+91) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e+47:
		tmp = -t1 / ((t1 + u) / (v / u))
	elif u <= 4.9e+91:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+47)
		tmp = Float64(Float64(-t1) / Float64(Float64(t1 + u) / Float64(v / u)));
	elseif (u <= 4.9e+91)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e+47)
		tmp = -t1 / ((t1 + u) / (v / u));
	elseif (u <= 4.9e+91)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+47], N[((-t1) / N[(N[(t1 + u), $MachinePrecision] / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.9e+91], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.1499999999999999e47

   1. Initial program 77.2%

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

      1. associate-/l* 79.4%

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

      2. associate-/l* 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in t1 around 0 93.3%

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

   if -1.1499999999999999e47 < u < 4.9000000000000003e91

   1. Initial program 69.7%

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

      1. times-frac 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t1 around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

   if 4.9000000000000003e91 < u

   1. Initial program 81.7%

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

      1. times-frac 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in t1 around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. neg-mul-1 77.7%

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

      3. unpow2 77.7%

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

   6. Simplified 77.7%

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

      1. distribute-rgt-neg-in 77.7%

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

      2. times-frac 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. *-commutative 84.1%

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

      2. frac-2neg 84.1%

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

      3. remove-double-neg 84.1%

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

      4. associate-*l/ 82.0%

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

      5. add-sqr-sqrt 39.1%

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

      6. sqrt-unprod 56.8%

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

      7. sqr-neg 56.8%

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

      8. sqrt-unprod 32.8%

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

      9. add-sqr-sqrt 64.1%

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

      10. *-commutative 64.1%

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

      11. associate-*l/ 63.9%

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

      12. associate-*r/ 66.0%

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

      13. div-inv 66.0%

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

      14. add-sqr-sqrt 34.8%

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

      15. sqrt-unprod 59.0%

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

      16. sqr-neg 59.0%

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

      17. sqrt-unprod 42.9%

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

      18. add-sqr-sqrt 88.0%

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

      19. div-inv 88.0%

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+47}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 4.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+55} \lor \neg \left(u \leq 4.5 \cdot 10^{+150}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e+55) (not (<= u 4.5e+150)))
   (* t1 (/ v (* u u)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+55) || !(u <= 4.5e+150)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.3d+55)) .or. (.not. (u <= 4.5d+150))) then
        tmp = t1 * (v / (u * u))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+55) || !(u <= 4.5e+150)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e+55) or not (u <= 4.5e+150):
		tmp = t1 * (v / (u * u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e+55) || !(u <= 4.5e+150))
		tmp = Float64(t1 * Float64(v / Float64(u * 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 <= -2.3e+55) || ~((u <= 4.5e+150)))
		tmp = t1 * (v / (u * u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e+55], N[Not[LessEqual[u, 4.5e+150]], $MachinePrecision]], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.29999999999999987e55 or 4.5e150 < u

   1. Initial program 77.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 0 74.4%

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

      1. associate-*r/ 74.4%

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

      2. neg-mul-1 74.4%

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

      3. unpow2 74.4%

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

   6. Simplified 74.4%

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

      1. distribute-lft-neg-in 74.4%

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

      2. times-frac 89.8%

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

      3. add-sqr-sqrt 40.6%

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

      4. sqrt-unprod 66.2%

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

      5. sqr-neg 66.2%

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

      6. sqrt-unprod 38.5%

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

      7. add-sqr-sqrt 70.0%

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

   8. Applied egg-rr 70.0%

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

      1. associate-*l/ 70.0%

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

      2. associate-*r/ 71.2%

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

      3. associate-/r* 71.5%

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

   10. Simplified 71.5%

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

   if -2.29999999999999987e55 < u < 4.5e150

   1. Initial program 71.3%

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

      1. times-frac 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in t1 around inf 77.1%

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

      1. associate-*r/ 77.1%

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

      2. sub-neg 77.1%

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

      3. metadata-eval 77.1%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in u around 0 79.5%

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

      1. neg-mul-1 79.5%

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

   9. Simplified 79.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+55} \lor \neg \left(u \leq 4.5 \cdot 10^{+150}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) / (-1.0 - (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 / (t1 + u)) / ((-1.0d0) - (u / t1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. *-commutative 73.4%

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

   2. times-frac 98.2%

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

   3. neg-mul-1 98.2%

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

   4. associate-/l* 98.0%

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

   5. associate-*r/ 98.1%

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

   6. associate-/l* 98.1%

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

   7. associate-/l/ 98.1%

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

   8. neg-mul-1 98.1%

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

   9. *-lft-identity 98.1%

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

   10. metadata-eval 98.1%

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

   11. times-frac 98.1%

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

   12. neg-mul-1 98.1%

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

   13. remove-double-neg 98.1%

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

   14. neg-mul-1 98.1%

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

   15. sub0-neg 98.1%

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

   16. associate--r+ 98.1%

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

   17. neg-sub0 98.1%

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

   18. div-sub 98.1%

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

   19. distribute-frac-neg 98.1%

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

   20. *-inverses 98.1%

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

   21. metadata-eval 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 7: 59.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+144} \lor \neg \left(u \leq 3.05 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.5e+144) (not (<= u 3.05e+184))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.5e+144) || !(u <= 3.05e+184)) {
		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 <= (-8.5d+144)) .or. (.not. (u <= 3.05d+184))) 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 <= -8.5e+144) || !(u <= 3.05e+184)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.5e+144) or not (u <= 3.05e+184):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.5e+144) || !(u <= 3.05e+184))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.5e+144) || ~((u <= 3.05e+184)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.5e+144], N[Not[LessEqual[u, 3.05e+184]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.4999999999999998e144 or 3.05000000000000004e184 < u

   1. Initial program 75.2%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around 0 93.2%

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

      1. mul-1-neg 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in t1 around inf 36.7%

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

      1. mul-1-neg 36.7%

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

      2. distribute-frac-neg 36.7%

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

   9. Simplified 36.7%

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

   if -8.4999999999999998e144 < u < 3.05000000000000004e184

   1. Initial program 72.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 73.0%

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

      1. associate-*r/ 73.0%

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

      2. neg-mul-1 73.0%

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

   6. Simplified 73.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+144} \lor \neg \left(u \leq 3.05 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 59.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+181}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.3e+92)
   (/ v (+ t1 u))
   (if (<= u 4.8e+181) (/ (- v) t1) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e+92) {
		tmp = v / (t1 + u);
	} else if (u <= 4.8e+181) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.3d+92)) then
        tmp = v / (t1 + u)
    else if (u <= 4.8d+181) then
        tmp = -v / t1
    else
        tmp = -v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e+92) {
		tmp = v / (t1 + u);
	} else if (u <= 4.8e+181) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.3e+92:
		tmp = v / (t1 + u)
	elif u <= 4.8e+181:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.3e+92)
		tmp = Float64(v / Float64(t1 + u));
	elseif (u <= 4.8e+181)
		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 <= -2.3e+92)
		tmp = v / (t1 + u);
	elseif (u <= 4.8e+181)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.3e+92], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.8e+181], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.29999999999999998e92

   1. Initial program 74.3%

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

      1. times-frac 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t1 around inf 29.1%

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

      1. associate-*r/ 27.1%

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

      2. sub-neg 27.1%

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

      3. metadata-eval 27.1%

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

   6. Applied egg-rr 27.1%

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

   7. Taylor expanded in u around 0 40.1%

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

      1. neg-mul-1 40.1%

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

   9. Simplified 40.1%

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

       1. expm1-log1p-u 39.7%

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

       2. expm1-udef 62.4%

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

       3. add-sqr-sqrt 21.6%

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

       4. sqrt-unprod 59.8%

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

       5. sqr-neg 59.8%

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

       6. sqrt-unprod 40.7%

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

       7. add-sqr-sqrt 62.9%

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

   11. Applied egg-rr 62.9%

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

       1. expm1-def 36.9%

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

       2. expm1-log1p 37.0%

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

   13. Simplified 37.0%

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

   if -2.29999999999999998e92 < u < 4.80000000000000004e181

   1. Initial program 71.7%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in t1 around inf 75.1%

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

      1. associate-*r/ 75.1%

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

      2. neg-mul-1 75.1%

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

   6. Simplified 75.1%

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

   if 4.80000000000000004e181 < u

   1. Initial program 81.9%

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

      1. 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 0 94.0%

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

      1. mul-1-neg 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in t1 around inf 37.8%

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

      1. mul-1-neg 37.8%

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

      2. distribute-frac-neg 37.8%

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

   9. Simplified 37.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+181}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 9: 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 73.4%

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

   1. times-frac 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in t1 around inf 60.0%

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

   1. associate-*r/ 59.0%

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

   2. sub-neg 59.0%

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

   3. metadata-eval 59.0%

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

6. Applied egg-rr 59.0%

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

7. Taylor expanded in u around 0 66.0%

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

   1. neg-mul-1 66.0%

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

9. Simplified 66.0%

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

10. Final simplification 66.0%

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

Alternative 10: 54.5% accurate, 3.0× 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(Float64(-v) / t1)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. times-frac 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in t1 around inf 59.8%

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

   1. associate-*r/ 59.8%

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

   2. neg-mul-1 59.8%

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

6. Simplified 59.8%

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

7. Final simplification 59.8%

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

Alternative 11: 14.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. times-frac 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in t1 around inf 60.0%

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

5. Taylor expanded in u around inf 13.5%

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

6. Final simplification 13.5%

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

Reproduce

herbie shell --seed 2023224 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))