Rosa's DopplerBench

Percentage Accurate: 72.6% → 97.8%

Time: 10.2s

Alternatives: 13

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% 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: 97.8% accurate, 0.8× speedup

Math

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

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

Derivation

1. Initial program 76.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.8

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

4. Applied rewrites 97.8%

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

5. Final simplification 97.8%

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

Alternative 2: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u} \cdot \left(1 - \frac{u}{t1}\right)\\ \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (* (/ v (+ t1 u)) (- 1.0 (/ u t1))))))
   (if (<= t1 -4.7e+154)
     t_1
     (if (<= t1 4.5e+156) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -((v / (t1 + u)) * (1.0 - (u / t1)));
	double tmp;
	if (t1 <= -4.7e+154) {
		tmp = t_1;
	} else if (t1 <= 4.5e+156) {
		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 / (t1 + u)) * (1.0d0 - (u / t1)))
    if (t1 <= (-4.7d+154)) then
        tmp = t_1
    else if (t1 <= 4.5d+156) 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 / (t1 + u)) * (1.0 - (u / t1)));
	double tmp;
	if (t1 <= -4.7e+154) {
		tmp = t_1;
	} else if (t1 <= 4.5e+156) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -((v / (t1 + u)) * (1.0 - (u / t1)))
	tmp = 0
	if t1 <= -4.7e+154:
		tmp = t_1
	elif t1 <= 4.5e+156:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(Float64(v / Float64(t1 + u)) * Float64(1.0 - Float64(u / t1))))
	tmp = 0.0
	if (t1 <= -4.7e+154)
		tmp = t_1;
	elseif (t1 <= 4.5e+156)
		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 / (t1 + u)) * (1.0 - (u / t1)));
	tmp = 0.0;
	if (t1 <= -4.7e+154)
		tmp = t_1;
	elseif (t1 <= 4.5e+156)
		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[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -4.7e+154], t$95$1, If[LessEqual[t1, 4.5e+156], 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 < -4.69999999999999983e154 or 4.50000000000000031e156 < 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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if -4.69999999999999983e154 < t1 < 4.50000000000000031e156

   1. Initial program 87.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 89.3

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

   4. Applied rewrites 89.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;-\frac{v}{t1 + u} \cdot \left(1 - \frac{u}{t1}\right)\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u} \cdot \left(1 - \frac{u}{t1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - \frac{u}{t1}\right) \cdot \left(-\frac{v}{t1}\right)\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\mathsf{fma}\left(u, \frac{2}{t1}, -1\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.7e+154)
   (* (- 1.0 (/ u t1)) (- (/ v t1)))
   (if (<= t1 4.5e+156)
     (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))
     (* v (/ (fma u (/ 2.0 t1) -1.0) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.7e+154) {
		tmp = (1.0 - (u / t1)) * -(v / t1);
	} else if (t1 <= 4.5e+156) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = v * (fma(u, (2.0 / t1), -1.0) / t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.7e+154)
		tmp = Float64(Float64(1.0 - Float64(u / t1)) * Float64(-Float64(v / t1)));
	elseif (t1 <= 4.5e+156)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(v * Float64(fma(u, Float64(2.0 / t1), -1.0) / t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.7e+154], N[(N[(1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] * (-N[(v / t1), $MachinePrecision])), $MachinePrecision], If[LessEqual[t1, 4.5e+156], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(u * N[(2.0 / t1), $MachinePrecision] + -1.0), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.69999999999999983e154

   1. Initial program 47.2%

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

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{t1}\right)} \cdot \frac{t1}{t1 + u} \]
   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 92.4

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 92.6

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

   10. Applied rewrites 92.6%

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

   if -4.69999999999999983e154 < t1 < 4.50000000000000031e156

   1. Initial program 87.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 89.3

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

   4. Applied rewrites 89.3%

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

   if 4.50000000000000031e156 < t1

   1. Initial program 43.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 45.4

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{u}{t1} + \left(\mathsf{neg}\left(1\right)\right)}}{t1} \cdot v \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot u}{t1}} + \left(\mathsf{neg}\left(1\right)\right)}{t1} \cdot v \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{u \cdot 2}}{t1} + \left(\mathsf{neg}\left(1\right)\right)}{t1} \cdot v \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{u \cdot \frac{2}{t1}} + \left(\mathsf{neg}\left(1\right)\right)}{t1} \cdot v \]

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u, \frac{2}{t1}, -1\right)}{t1}} \cdot v \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - \frac{u}{t1}\right) \cdot \left(-\frac{v}{t1}\right)\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\mathsf{fma}\left(u, \frac{2}{t1}, -1\right)}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \frac{u}{t1}\right) \cdot \left(-\frac{v}{t1}\right)\\ \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (/ u t1)) (- (/ v t1)))))
   (if (<= t1 -4.7e+154)
     t_1
     (if (<= t1 4.5e+156) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (1.0 - (u / t1)) * -(v / t1);
	double tmp;
	if (t1 <= -4.7e+154) {
		tmp = t_1;
	} else if (t1 <= 4.5e+156) {
		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 = (1.0d0 - (u / t1)) * -(v / t1)
    if (t1 <= (-4.7d+154)) then
        tmp = t_1
    else if (t1 <= 4.5d+156) 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 = (1.0 - (u / t1)) * -(v / t1);
	double tmp;
	if (t1 <= -4.7e+154) {
		tmp = t_1;
	} else if (t1 <= 4.5e+156) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (1.0 - (u / t1)) * -(v / t1)
	tmp = 0
	if t1 <= -4.7e+154:
		tmp = t_1
	elif t1 <= 4.5e+156:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(1.0 - Float64(u / t1)) * Float64(-Float64(v / t1)))
	tmp = 0.0
	if (t1 <= -4.7e+154)
		tmp = t_1;
	elseif (t1 <= 4.5e+156)
		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 = (1.0 - (u / t1)) * -(v / t1);
	tmp = 0.0;
	if (t1 <= -4.7e+154)
		tmp = t_1;
	elseif (t1 <= 4.5e+156)
		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[(N[(1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] * (-N[(v / t1), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t1, -4.7e+154], t$95$1, If[LessEqual[t1, 4.5e+156], 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 < -4.69999999999999983e154 or 4.50000000000000031e156 < 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 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 94.0

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

   10. Applied rewrites 94.0%

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

   if -4.69999999999999983e154 < t1 < 4.50000000000000031e156

   1. Initial program 87.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 89.3

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

   4. Applied rewrites 89.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - \frac{u}{t1}\right) \cdot \left(-\frac{v}{t1}\right)\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{u}{t1}\right) \cdot \left(-\frac{v}{t1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1}\\ \mathbf{if}\;t1 \leq -7 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v t1))))
   (if (<= t1 -7e+153)
     t_1
     (if (<= t1 4.5e+156) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / t1);
	double tmp;
	if (t1 <= -7e+153) {
		tmp = t_1;
	} else if (t1 <= 4.5e+156) {
		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 / t1)
    if (t1 <= (-7d+153)) then
        tmp = t_1
    else if (t1 <= 4.5d+156) 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 / t1);
	double tmp;
	if (t1 <= -7e+153) {
		tmp = t_1;
	} else if (t1 <= 4.5e+156) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / t1)
	tmp = 0
	if t1 <= -7e+153:
		tmp = t_1
	elif t1 <= 4.5e+156:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / t1))
	tmp = 0.0
	if (t1 <= -7e+153)
		tmp = t_1;
	elseif (t1 <= 4.5e+156)
		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 / t1);
	tmp = 0.0;
	if (t1 <= -7e+153)
		tmp = t_1;
	elseif (t1 <= 4.5e+156)
		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 / t1), $MachinePrecision])}, If[LessEqual[t1, -7e+153], t$95$1, If[LessEqual[t1, 4.5e+156], 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 < -6.9999999999999998e153 or 4.50000000000000031e156 < t1

   1. Initial program 45.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 92.7

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

   5. Applied rewrites 92.7%

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

   if -6.9999999999999998e153 < t1 < 4.50000000000000031e156

   1. Initial program 87.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 89.7

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

   4. Applied rewrites 89.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+153}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u} \cdot \left(-1\right)\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.12 \cdot 10^{-116}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (+ t1 u)) (- 1.0))))
   (if (<= t1 -4e-9) t_1 (if (<= t1 1.12e-116) (* 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 <= -4e-9) {
		tmp = t_1;
	} else if (t1 <= 1.12e-116) {
		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 <= (-4d-9)) then
        tmp = t_1
    else if (t1 <= 1.12d-116) 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 <= -4e-9) {
		tmp = t_1;
	} else if (t1 <= 1.12e-116) {
		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 <= -4e-9:
		tmp = t_1
	elif t1 <= 1.12e-116:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) * Float64(-1.0))
	tmp = 0.0
	if (t1 <= -4e-9)
		tmp = t_1;
	elseif (t1 <= 1.12e-116)
		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 <= -4e-9)
		tmp = t_1;
	elseif (t1 <= 1.12e-116)
		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, -4e-9], t$95$1, If[LessEqual[t1, 1.12e-116], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.00000000000000025e-9 or 1.12e-116 < t1

   1. Initial program 70.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 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 82.8%

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

   if -4.00000000000000025e-9 < t1 < 1.12e-116

   1. Initial program 85.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 79.2

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

   5. Applied rewrites 79.2%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. 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 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1\right)\\ \mathbf{elif}\;t1 \leq 1.12 \cdot 10^{-116}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u} \cdot \left(-1\right)\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.12 \cdot 10^{-116}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (+ t1 u)) (- 1.0))))
   (if (<= t1 -4e-9) t_1 (if (<= t1 1.12e-116) (* 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 <= -4e-9) {
		tmp = t_1;
	} else if (t1 <= 1.12e-116) {
		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 <= (-4d-9)) then
        tmp = t_1
    else if (t1 <= 1.12d-116) 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 <= -4e-9) {
		tmp = t_1;
	} else if (t1 <= 1.12e-116) {
		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 <= -4e-9:
		tmp = t_1
	elif t1 <= 1.12e-116:
		tmp = t1 * (v / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 + u)) * Float64(-1.0))
	tmp = 0.0
	if (t1 <= -4e-9)
		tmp = t_1;
	elseif (t1 <= 1.12e-116)
		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 <= -4e-9)
		tmp = t_1;
	elseif (t1 <= 1.12e-116)
		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, -4e-9], t$95$1, If[LessEqual[t1, 1.12e-116], N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.00000000000000025e-9 or 1.12e-116 < t1

   1. Initial program 70.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 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 82.8%

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

   if -4.00000000000000025e-9 < t1 < 1.12e-116

   1. Initial program 85.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1\right)\\ \mathbf{elif}\;t1 \leq 1.12 \cdot 10^{-116}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.8

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

4. Applied rewrites 97.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 97.6

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

6. Applied rewrites 97.6%

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

7. Final simplification 97.6%

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

Alternative 9: 97.9% accurate, 0.8× speedup

Math

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

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(-Float64(t1 / 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 76.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.6

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

4. Applied rewrites 97.6%

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

5. Final simplification 97.6%

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

Alternative 10: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v \cdot 1}{-u}\\ \mathbf{if}\;u \leq -5.6 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.25 \cdot 10^{+201}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* v 1.0) (- u))))
   (if (<= u -5.6e+106) t_1 (if (<= u 2.25e+201) (- (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v * 1.0) / -u;
	double tmp;
	if (u <= -5.6e+106) {
		tmp = t_1;
	} else if (u <= 2.25e+201) {
		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 * 1.0d0) / -u
    if (u <= (-5.6d+106)) then
        tmp = t_1
    else if (u <= 2.25d+201) 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 * 1.0) / -u;
	double tmp;
	if (u <= -5.6e+106) {
		tmp = t_1;
	} else if (u <= 2.25e+201) {
		tmp = -(v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v * 1.0) / -u
	tmp = 0
	if u <= -5.6e+106:
		tmp = t_1
	elif u <= 2.25e+201:
		tmp = -(v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v * 1.0) / Float64(-u))
	tmp = 0.0
	if (u <= -5.6e+106)
		tmp = t_1;
	elseif (u <= 2.25e+201)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v * 1.0) / -u;
	tmp = 0.0;
	if (u <= -5.6e+106)
		tmp = t_1;
	elseif (u <= 2.25e+201)
		tmp = -(v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v * 1.0), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -5.6e+106], t$95$1, If[LessEqual[u, 2.25e+201], (-N[(v / t1), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.59999999999999986e106 or 2.25000000000000005e201 < u

   1. Initial program 79.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

   7. Taylor expanded in t1 around inf

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

   9. Applied rewrites 53.6%

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

   10. Taylor expanded in t1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 45.0

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

   12. Applied rewrites 45.0%

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

   if -5.59999999999999986e106 < u < 2.25000000000000005e201

   1. Initial program 75.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{v \cdot 1}{-u}\\ \mathbf{elif}\;u \leq 2.25 \cdot 10^{+201}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot 1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.5% accurate, 1.4× speedup

Math

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

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(t1 + u)) * Float64(-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 76.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.6

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

4. Applied rewrites 97.6%

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

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 62.3%

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

8. Final simplification 62.3%

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

Alternative 12: 61.3% accurate, 1.4× speedup

Math

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

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(Float64(-v) * Float64(1.0 / 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 76.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.8

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

4. Applied rewrites 97.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 97.6

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in t1 around inf

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

9. Applied rewrites 62.3%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-neg.f64 N/A

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

    4. neg-mul-1 N/A

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

    5. *-commutative N/A

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

    6. times-frac N/A

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

    7. div-inv N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. neg-mul-1 N/A

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

    11. lift-neg.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-/.f64 62.2

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

11. Applied rewrites 62.2%

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

12. Final simplification 62.2%

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

Alternative 13: 53.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 54.2

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

5. Applied rewrites 54.2%

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

6. Final simplification 54.2%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))