Rosa's DopplerBench

Percentage Accurate: 73.4% → 98.1%

Time: 11.2s

Alternatives: 12

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: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)))
   (if (<= t1 -1.08e+159)
     (/ v (- t1))
     (if (<= t1 2e+128)
       (* v (/ t1 (* (+ t1 u) t_1)))
       (/ (* t1 (/ v t1)) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = v / -t1;
	} else if (t1 <= 2e+128) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = (t1 * (v / t1)) / 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 = -t1 - u
    if (t1 <= (-1.08d+159)) then
        tmp = v / -t1
    else if (t1 <= 2d+128) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = (t1 * (v / t1)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = v / -t1;
	} else if (t1 <= 2e+128) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = (t1 * (v / t1)) / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 - u
	tmp = 0
	if t1 <= -1.08e+159:
		tmp = v / -t1
	elif t1 <= 2e+128:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = (t1 * (v / t1)) / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	tmp = 0.0
	if (t1 <= -1.08e+159)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 2e+128)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = Float64(Float64(t1 * Float64(v / t1)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 - u;
	tmp = 0.0;
	if (t1 <= -1.08e+159)
		tmp = v / -t1;
	elseif (t1 <= 2e+128)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = (t1 * (v / t1)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, If[LessEqual[t1, -1.08e+159], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 2e+128], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / t1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.07999999999999991e159

   1. Initial program 47.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.07999999999999991e159 < t1 < 2.0000000000000002e128

   1. Initial program 80.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 87.1

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

   4. Applied rewrites 87.1%

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

   if 2.0000000000000002e128 < t1

   1. Initial program 45.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

      1. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1}}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{-t1}\\ \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- t1))))
   (if (<= t1 -1.08e+159)
     t_1
     (if (<= t1 2e+128)
       (* v (/ t1 (* (+ t1 u) (- (- t1) u))))
       (* (/ t1 (+ t1 u)) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = v / -t1;
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = t_1;
	} else if (t1 <= 2e+128) {
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)));
	} else {
		tmp = (t1 / (t1 + u)) * 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 / -t1
    if (t1 <= (-1.08d+159)) then
        tmp = t_1
    else if (t1 <= 2d+128) then
        tmp = v * (t1 / ((t1 + u) * (-t1 - u)))
    else
        tmp = (t1 / (t1 + u)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / -t1;
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = t_1;
	} else if (t1 <= 2e+128) {
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)));
	} else {
		tmp = (t1 / (t1 + u)) * t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / -t1
	tmp = 0
	if t1 <= -1.08e+159:
		tmp = t_1
	elif t1 <= 2e+128:
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)))
	else:
		tmp = (t1 / (t1 + u)) * t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(-t1))
	tmp = 0.0
	if (t1 <= -1.08e+159)
		tmp = t_1;
	elseif (t1 <= 2e+128)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-t1) - u))));
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / -t1;
	tmp = 0.0;
	if (t1 <= -1.08e+159)
		tmp = t_1;
	elseif (t1 <= 2e+128)
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)));
	else
		tmp = (t1 / (t1 + u)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / (-t1)), $MachinePrecision]}, If[LessEqual[t1, -1.08e+159], t$95$1, If[LessEqual[t1, 2e+128], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.07999999999999991e159

   1. Initial program 47.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.07999999999999991e159 < t1 < 2.0000000000000002e128

   1. Initial program 80.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 87.1

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

   4. Applied rewrites 87.1%

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

   if 2.0000000000000002e128 < t1

   1. Initial program 45.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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-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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)))
   (if (<= t1 -1.08e+159)
     (/ v (- t1))
     (if (<= t1 2e+128) (* v (/ t1 (* (+ t1 u) t_1))) (* (/ v t_1) 1.0)))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = v / -t1;
	} else if (t1 <= 2e+128) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = (v / t_1) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 - u
    if (t1 <= (-1.08d+159)) then
        tmp = v / -t1
    else if (t1 <= 2d+128) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = (v / t_1) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (t1 <= -1.08e+159) {
		tmp = v / -t1;
	} else if (t1 <= 2e+128) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = (v / t_1) * 1.0;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 - u
	tmp = 0
	if t1 <= -1.08e+159:
		tmp = v / -t1
	elif t1 <= 2e+128:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = (v / t_1) * 1.0
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	tmp = 0.0
	if (t1 <= -1.08e+159)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 2e+128)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = Float64(Float64(v / t_1) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 - u;
	tmp = 0.0;
	if (t1 <= -1.08e+159)
		tmp = v / -t1;
	elseif (t1 <= 2e+128)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = (v / t_1) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, If[LessEqual[t1, -1.08e+159], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 2e+128], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.07999999999999991e159

   1. Initial program 47.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.07999999999999991e159 < t1 < 2.0000000000000002e128

   1. Initial program 80.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 87.1

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

   4. Applied rewrites 87.1%

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

   if 2.0000000000000002e128 < t1

   1. Initial program 45.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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-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 99.9

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

   4. Applied rewrites 99.9%

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

   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 92.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+128}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u} \cdot 1\\ \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (- (- t1) u)) 1.0)))
   (if (<= t1 -1.9e-42)
     t_1
     (if (<= t1 1.42e-12) (* (/ v (- u)) (/ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (-t1 - u)) * 1.0;
	double tmp;
	if (t1 <= -1.9e-42) {
		tmp = t_1;
	} else if (t1 <= 1.42e-12) {
		tmp = (v / -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 / (-t1 - u)) * 1.0d0
    if (t1 <= (-1.9d-42)) then
        tmp = t_1
    else if (t1 <= 1.42d-12) then
        tmp = (v / -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 / (-t1 - u)) * 1.0;
	double tmp;
	if (t1 <= -1.9e-42) {
		tmp = t_1;
	} else if (t1 <= 1.42e-12) {
		tmp = (v / -u) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (-t1 - u)) * 1.0
	tmp = 0
	if t1 <= -1.9e-42:
		tmp = t_1
	elif t1 <= 1.42e-12:
		tmp = (v / -u) * (t1 / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(Float64(-t1) - u)) * 1.0)
	tmp = 0.0
	if (t1 <= -1.9e-42)
		tmp = t_1;
	elseif (t1 <= 1.42e-12)
		tmp = Float64(Float64(v / Float64(-u)) * Float64(t1 / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / (-t1 - u)) * 1.0;
	tmp = 0.0;
	if (t1 <= -1.9e-42)
		tmp = t_1;
	elseif (t1 <= 1.42e-12)
		tmp = (v / -u) * (t1 / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t1, -1.9e-42], t$95$1, If[LessEqual[t1, 1.42e-12], N[(N[(v / (-u)), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.90000000000000009e-42 or 1.42e-12 < t1

   1. Initial program 59.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 85.6%

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

   if -1.90000000000000009e-42 < t1 < 1.42e-12

   1. Initial program 80.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in t1 around 0

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

      1. lower-/.f64 75.7

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in t1 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 77.8

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

   10. Applied rewrites 77.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-42}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u} \cdot 1\\ \mathbf{if}\;t1 \leq -1.85 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (- (- t1) u)) 1.0)))
   (if (<= t1 -1.85e-117)
     t_1
     (if (<= t1 1.42e-12) (* v (/ (- t1) (* u u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (-t1 - u)) * 1.0;
	double tmp;
	if (t1 <= -1.85e-117) {
		tmp = t_1;
	} else if (t1 <= 1.42e-12) {
		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 / (-t1 - u)) * 1.0d0
    if (t1 <= (-1.85d-117)) then
        tmp = t_1
    else if (t1 <= 1.42d-12) 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 / (-t1 - u)) * 1.0;
	double tmp;
	if (t1 <= -1.85e-117) {
		tmp = t_1;
	} else if (t1 <= 1.42e-12) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (-t1 - u)) * 1.0
	tmp = 0
	if t1 <= -1.85e-117:
		tmp = t_1
	elif t1 <= 1.42e-12:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(Float64(-t1) - u)) * 1.0)
	tmp = 0.0
	if (t1 <= -1.85e-117)
		tmp = t_1;
	elseif (t1 <= 1.42e-12)
		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 / (-t1 - u)) * 1.0;
	tmp = 0.0;
	if (t1 <= -1.85e-117)
		tmp = t_1;
	elseif (t1 <= 1.42e-12)
		tmp = v * (-t1 / (u * u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t1, -1.85e-117], t$95$1, If[LessEqual[t1, 1.42e-12], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.8500000000000001e-117 or 1.42e-12 < t1

   1. Initial program 60.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 81.4%

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

   if -1.8500000000000001e-117 < t1 < 1.42e-12

   1. Initial program 82.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 77.4

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

   5. Applied rewrites 77.4%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
   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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 81.5

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

   7. Applied rewrites 81.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{-117}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u} \cdot 1\\ \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (- (- t1) u)) 1.0)))
   (if (<= t1 -1.9e-42)
     t_1
     (if (<= t1 1.42e-12) (* t1 (/ v (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (-t1 - u)) * 1.0;
	double tmp;
	if (t1 <= -1.9e-42) {
		tmp = t_1;
	} else if (t1 <= 1.42e-12) {
		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 / (-t1 - u)) * 1.0d0
    if (t1 <= (-1.9d-42)) then
        tmp = t_1
    else if (t1 <= 1.42d-12) 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 / (-t1 - u)) * 1.0;
	double tmp;
	if (t1 <= -1.9e-42) {
		tmp = t_1;
	} else if (t1 <= 1.42e-12) {
		tmp = t1 * (v / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (-t1 - u)) * 1.0
	tmp = 0
	if t1 <= -1.9e-42:
		tmp = t_1
	elif t1 <= 1.42e-12:
		tmp = t1 * (v / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(Float64(-t1) - u)) * 1.0)
	tmp = 0.0
	if (t1 <= -1.9e-42)
		tmp = t_1;
	elseif (t1 <= 1.42e-12)
		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 / (-t1 - u)) * 1.0;
	tmp = 0.0;
	if (t1 <= -1.9e-42)
		tmp = t_1;
	elseif (t1 <= 1.42e-12)
		tmp = t1 * (v / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t1, -1.9e-42], t$95$1, If[LessEqual[t1, 1.42e-12], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.90000000000000009e-42 or 1.42e-12 < t1

   1. Initial program 59.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 85.6%

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

   if -1.90000000000000009e-42 < t1 < 1.42e-12

   1. Initial program 80.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-42}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 9: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \frac{v}{-u}\\ \mathbf{if}\;u \leq -1.75 \cdot 10^{+223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 7 \cdot 10^{+218}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* 1.0 (/ v (- u)))))
   (if (<= u -1.75e+223) t_1 (if (<= u 7e+218) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = 1.0 * (v / -u);
	double tmp;
	if (u <= -1.75e+223) {
		tmp = t_1;
	} else if (u <= 7e+218) {
		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 = 1.0d0 * (v / -u)
    if (u <= (-1.75d+223)) then
        tmp = t_1
    else if (u <= 7d+218) 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 = 1.0 * (v / -u);
	double tmp;
	if (u <= -1.75e+223) {
		tmp = t_1;
	} else if (u <= 7e+218) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = 1.0 * (v / -u)
	tmp = 0
	if u <= -1.75e+223:
		tmp = t_1
	elif u <= 7e+218:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(1.0 * Float64(v / Float64(-u)))
	tmp = 0.0
	if (u <= -1.75e+223)
		tmp = t_1;
	elseif (u <= 7e+218)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = 1.0 * (v / -u);
	tmp = 0.0;
	if (u <= -1.75e+223)
		tmp = t_1;
	elseif (u <= 7e+218)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(1.0 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.75e+223], t$95$1, If[LessEqual[u, 7e+218], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.75000000000000005e223 or 7.00000000000000038e218 < u

   1. Initial program 78.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   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 60.6%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 58.4

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

   10. Applied rewrites 58.4%

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

   if -1.75000000000000005e223 < u < 7.00000000000000038e218

   1. Initial program 66.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 65.6

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

   5. Applied rewrites 65.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+223}:\\ \;\;\;\;1 \cdot \frac{v}{-u}\\ \mathbf{elif}\;u \leq 7 \cdot 10^{+218}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (-t1 - u)) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.1

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

4. Applied rewrites 98.1%

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

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 66.6%

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

8. Final simplification 66.6%

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

Alternative 11: 61.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

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

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

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

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 73.4

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

4. Applied rewrites 73.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. lower-/.f64 94.8

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

6. Applied rewrites 94.8%

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

7. Taylor expanded in t1 around inf

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

9. Applied rewrites 66.4%

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

10. Final simplification 66.4%

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

Alternative 12: 54.0% 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 68.1%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 58.3

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

5. Applied rewrites 58.3%

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

6. Final simplification 58.3%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))