Rosa's DopplerBench

Percentage Accurate: 72.1% → 97.9%

Time: 6.8s

Alternatives: 7

Speedup: 0.8×

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: 72.1% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 72.3

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 98.6%

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

Alternative 2: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{+120}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.8e+25)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 1.1e+120)
     (* (/ (- v) (* (+ u t1) (+ u t1))) t1)
     (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.8e+25) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.1e+120) {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.8d+25)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 1.1d+120) then
        tmp = (-v / ((u + t1) * (u + t1))) * t1
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.8e+25) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.1e+120) {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.8e+25:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 1.1e+120:
		tmp = (-v / ((u + t1) * (u + t1))) * t1
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.8e+25)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 1.1e+120)
		tmp = Float64(Float64(Float64(-v) / Float64(Float64(u + t1) * Float64(u + t1))) * t1);
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.8e+25)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 1.1e+120)
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.8e+25], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.1e+120], N[(N[((-v) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.7999999999999998e25

   1. Initial program 49.6%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 96.5%

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

   if -5.7999999999999998e25 < t1 < 1.1000000000000001e120

   1. Initial program 83.8%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

   if 1.1000000000000001e120 < t1

   1. Initial program 38.7%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 42.6

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

   5. Applied rewrites 42.6%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 92.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{+120}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 32000000000000:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.75e-62)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 32000000000000.0) (* t1 (/ (/ (- v) u) u)) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.75e-62) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 32000000000000.0) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.75d-62)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 32000000000000.0d0) then
        tmp = t1 * ((-v / u) / u)
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.75e-62) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 32000000000000.0) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.75e-62:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 32000000000000.0:
		tmp = t1 * ((-v / u) / u)
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.75e-62)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 32000000000000.0)
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / u));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.75e-62)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 32000000000000.0)
		tmp = t1 * ((-v / u) / u);
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.75e-62], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 32000000000000.0], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.7500000000000001e-62

   1. Initial program 62.8%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 89.5%

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

   if -1.7500000000000001e-62 < t1 < 3.2e13

   1. Initial program 79.8%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 77.6%

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

   9. Applied rewrites 79.7%

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

   if 3.2e13 < t1

   1. Initial program 58.1%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 91.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 32000000000000:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.9e-63)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 5.6e-136) (* (/ (- v) (* u u)) t1) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9e-63) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 5.6e-136) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.9d-63)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 5.6d-136) then
        tmp = (-v / (u * u)) * t1
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9e-63) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 5.6e-136) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.9e-63:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 5.6e-136:
		tmp = (-v / (u * u)) * t1
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.9e-63)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 5.6e-136)
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.9e-63)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 5.6e-136)
		tmp = (-v / (u * u)) * t1;
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.9e-63], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.6e-136], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.90000000000000009e-63

   1. Initial program 62.6%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 88.5%

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

   if -1.90000000000000009e-63 < t1 < 5.6000000000000002e-136

   1. Initial program 79.8%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 80.9%

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

   if 5.6000000000000002e-136 < t1

   1. Initial program 63.4%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 83.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-136}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.9e-63)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 5.6e-136) (* (- v) (/ t1 (* u u))) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9e-63) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 5.6e-136) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.9d-63)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 5.6d-136) then
        tmp = -v * (t1 / (u * u))
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9e-63) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 5.6e-136) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.9e-63:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 5.6e-136:
		tmp = -v * (t1 / (u * u))
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.9e-63)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 5.6e-136)
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.9e-63)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 5.6e-136)
		tmp = -v * (t1 / (u * u));
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.9e-63], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.6e-136], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.90000000000000009e-63

   1. Initial program 62.6%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 88.5%

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

   if -1.90000000000000009e-63 < t1 < 5.6000000000000002e-136

   1. Initial program 79.8%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 79.5%

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

   if 5.6000000000000002e-136 < t1

   1. Initial program 63.4%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 83.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-136}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% accurate, 1.8× 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(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 69.5%

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

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{\left(t1 + u\right)}^{2}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 72.3

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 66.2%

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

Alternative 7: 54.2% accurate, 2.1× 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 69.5%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

   4. lower-neg.f64 56.8

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

5. Applied rewrites 56.8%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))