Rosa's DopplerBench

Percentage Accurate: 72.7% → 97.9%

Time: 9.8s

Alternatives: 10

Speedup: 0.9×

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: 72.7% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -2.6e+158)
     t_1
     (if (<= t1 6.8e+140) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.6e+158) {
		tmp = t_1;
	} else if (t1 <= 6.8e+140) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-2.6d+158)) then
        tmp = t_1
    else if (t1 <= 6.8d+140) then
        tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.6e+158) {
		tmp = t_1;
	} else if (t1 <= 6.8e+140) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2.6e+158:
		tmp = t_1
	elif t1 <= 6.8e+140:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -2.6e+158)
		tmp = t_1;
	elseif (t1 <= 6.8e+140)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2.6e+158)
		tmp = t_1;
	elseif (t1 <= 6.8e+140)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.6e+158], t$95$1, If[LessEqual[t1, 6.8e+140], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.6e158 or 6.8e140 < t1

   1. Initial program 28.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t1 around 0

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.1

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

   9. Applied rewrites 94.1%

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

   10. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 85.8

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

   12. Applied rewrites 85.8%

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

   if -2.6e158 < t1 < 6.8e140

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.7

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

   4. Applied rewrites 90.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.8× 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 69.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in t1 around 0

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

   1. div-sub N/A

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

   2. associate-/l* N/A

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

   3. *-inverses N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 98.0

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

7. Applied rewrites 98.0%

   \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{-1 - \frac{u}{t1}}} \]
8. Add Preprocessing

Alternative 4: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -7.2 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -7.2e-79)
     t_1
     (if (<= t1 2.1e-35) (* v (/ (- t1) (* u u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -7.2e-79) {
		tmp = t_1;
	} else if (t1 <= 2.1e-35) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-7.2d-79)) then
        tmp = t_1
    else if (t1 <= 2.1d-35) then
        tmp = v * (-t1 / (u * u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -7.2e-79) {
		tmp = t_1;
	} else if (t1 <= 2.1e-35) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -7.2e-79:
		tmp = t_1
	elif t1 <= 2.1e-35:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -7.2e-79)
		tmp = t_1;
	elseif (t1 <= 2.1e-35)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -7.2e-79)
		tmp = t_1;
	elseif (t1 <= 2.1e-35)
		tmp = v * (-t1 / (u * u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.2e-79], t$95$1, If[LessEqual[t1, 2.1e-35], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.2000000000000005e-79 or 2.1e-35 < t1

   1. Initial program 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.5

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

   9. Applied rewrites 94.5%

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

   10. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 79.0

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

   12. Applied rewrites 79.0%

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

   if -7.2000000000000005e-79 < t1 < 2.1e-35

   1. Initial program 82.2%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 79.8

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

   7. Applied rewrites 79.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{-35}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -2.3e-80)
     t_1
     (if (<= t1 2.1e-35) (* t1 (/ v (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.3e-80) {
		tmp = t_1;
	} else if (t1 <= 2.1e-35) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-2.3d-80)) then
        tmp = t_1
    else if (t1 <= 2.1d-35) then
        tmp = t1 * (v / (u * -u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.3e-80) {
		tmp = t_1;
	} else if (t1 <= 2.1e-35) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2.3e-80:
		tmp = t_1
	elif t1 <= 2.1e-35:
		tmp = t1 * (v / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -2.3e-80)
		tmp = t_1;
	elseif (t1 <= 2.1e-35)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2.3e-80)
		tmp = t_1;
	elseif (t1 <= 2.1e-35)
		tmp = t1 * (v / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.3e-80], t$95$1, If[LessEqual[t1, 2.1e-35], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.2999999999999998e-80 or 2.1e-35 < t1

   1. Initial program 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.5

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

   9. Applied rewrites 94.5%

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

   10. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 79.0

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

   12. Applied rewrites 79.0%

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

   if -2.2999999999999998e-80 < t1 < 2.1e-35

   1. Initial program 82.2%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 75.0

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

   5. Applied rewrites 75.0%

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

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 7: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)} \end{array} \]

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(v / Float64(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[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. distribute-neg-in N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. lower--.f64 N/A

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

   23. lower-/.f64 95.1

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

7. Applied rewrites 95.1%

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

8. Final simplification 95.1%

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

Alternative 8: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u} \cdot 1\\ \mathbf{if}\;u \leq -2.1 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) u) 1.0)))
   (if (<= u -2.1e+153) t_1 (if (<= u 2e+160) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / u) * 1.0;
	double tmp;
	if (u <= -2.1e+153) {
		tmp = t_1;
	} else if (u <= 2e+160) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-v / u) * 1.0d0
    if (u <= (-2.1d+153)) then
        tmp = t_1
    else if (u <= 2d+160) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-v / u) * 1.0;
	double tmp;
	if (u <= -2.1e+153) {
		tmp = t_1;
	} else if (u <= 2e+160) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-v / u) * 1.0
	tmp = 0
	if u <= -2.1e+153:
		tmp = t_1
	elif u <= 2e+160:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-v) / u) * 1.0)
	tmp = 0.0
	if (u <= -2.1e+153)
		tmp = t_1;
	elseif (u <= 2e+160)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-v / u) * 1.0;
	tmp = 0.0;
	if (u <= -2.1e+153)
		tmp = t_1;
	elseif (u <= 2e+160)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / u), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[u, -2.1e+153], t$95$1, If[LessEqual[u, 2e+160], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.10000000000000017e153 or 2.00000000000000001e160 < u

   1. Initial program 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 35.6

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

   10. Applied rewrites 35.6%

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

   if -2.10000000000000017e153 < u < 2.00000000000000001e160

   1. Initial program 68.7%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 64.0

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

   5. Applied rewrites 64.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-v}{u} \cdot 1\\ \mathbf{elif}\;u \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in t1 around 0

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

   1. div-sub N/A

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

   2. associate-/l* N/A

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

   3. *-inverses N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 98.0

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

7. Applied rewrites 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 95.1

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

9. Applied rewrites 95.1%

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

10. Taylor expanded in u around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 60.0

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

12. Applied rewrites 60.0%

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

Alternative 10: 53.6% 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(v / Float64(-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 69.7%

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

3. Taylor expanded in t1 around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
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 51.2

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

5. Applied rewrites 51.2%

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

6. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))