Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 8.5s

Alternatives: 12

Speedup: 1.0×

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: 72.4% 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, 1.0× 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 69.9%

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

   1. associate-/r* 83.8%

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

   2. associate-/l* 96.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in t1 around 0 93.4%

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

   1. +-commutative 93.4%

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

6. Simplified 93.4%

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

7. Taylor expanded in v around -inf 69.9%

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

   1. times-frac 98.6%

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

   2. mul-1-neg 98.6%

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

   3. neg-mul-1 98.6%

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

   4. +-commutative 98.6%

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

   5. sub-neg 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 2: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -1.66 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.02 \cdot 10^{-184}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))))
        (t_2 (/ (- (/ u (/ t1 v)) v) (+ t1 u))))
   (if (<= t1 -1.6e+123)
     t_2
     (if (<= t1 -1.66e-102)
       t_1
       (if (<= t1 1.02e-184)
         (* (/ t1 u) (/ (- v) u))
         (if (<= t1 4.5e+148) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double t_2 = ((u / (t1 / v)) - v) / (t1 + u);
	double tmp;
	if (t1 <= -1.6e+123) {
		tmp = t_2;
	} else if (t1 <= -1.66e-102) {
		tmp = t_1;
	} else if (t1 <= 1.02e-184) {
		tmp = (t1 / u) * (-v / u);
	} else if (t1 <= 4.5e+148) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    t_2 = ((u / (t1 / v)) - v) / (t1 + u)
    if (t1 <= (-1.6d+123)) then
        tmp = t_2
    else if (t1 <= (-1.66d-102)) then
        tmp = t_1
    else if (t1 <= 1.02d-184) then
        tmp = (t1 / u) * (-v / u)
    else if (t1 <= 4.5d+148) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double t_2 = ((u / (t1 / v)) - v) / (t1 + u);
	double tmp;
	if (t1 <= -1.6e+123) {
		tmp = t_2;
	} else if (t1 <= -1.66e-102) {
		tmp = t_1;
	} else if (t1 <= 1.02e-184) {
		tmp = (t1 / u) * (-v / u);
	} else if (t1 <= 4.5e+148) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	t_2 = ((u / (t1 / v)) - v) / (t1 + u)
	tmp = 0
	if t1 <= -1.6e+123:
		tmp = t_2
	elif t1 <= -1.66e-102:
		tmp = t_1
	elif t1 <= 1.02e-184:
		tmp = (t1 / u) * (-v / u)
	elif t1 <= 4.5e+148:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	t_2 = Float64(Float64(Float64(u / Float64(t1 / v)) - v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.6e+123)
		tmp = t_2;
	elseif (t1 <= -1.66e-102)
		tmp = t_1;
	elseif (t1 <= 1.02e-184)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif (t1 <= 4.5e+148)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	t_2 = ((u / (t1 / v)) - v) / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1.6e+123)
		tmp = t_2;
	elseif (t1 <= -1.66e-102)
		tmp = t_1;
	elseif (t1 <= 1.02e-184)
		tmp = (t1 / u) * (-v / u);
	elseif (t1 <= 4.5e+148)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(u / N[(t1 / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.6e+123], t$95$2, If[LessEqual[t1, -1.66e-102], t$95$1, If[LessEqual[t1, 1.02e-184], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.5e+148], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.60000000000000002e123 or 4.49999999999999994e148 < t1

   1. Initial program 39.1%

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

      1. associate-/r* 69.5%

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

      2. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around inf 83.1%

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

      1. neg-mul-1 83.1%

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

      2. +-commutative 83.1%

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

      3. unsub-neg 83.1%

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

      4. associate-/l* 92.6%

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

   6. Simplified 92.6%

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

   if -1.60000000000000002e123 < t1 < -1.66000000000000001e-102 or 1.0200000000000001e-184 < t1 < 4.49999999999999994e148

   1. Initial program 85.6%

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

      1. associate-*l/ 91.0%

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

      2. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   if -1.66000000000000001e-102 < t1 < 1.0200000000000001e-184

   1. Initial program 75.9%

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

      1. associate-/r* 82.9%

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

      2. associate-/l* 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in v around 0 82.9%

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

      1. mul-1-neg 82.9%

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

      2. associate-*l/ 93.7%

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

      3. distribute-rgt-neg-out 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in t1 around 0 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 89.3%

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

      3. *-commutative 89.3%

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

      4. neg-mul-1 89.3%

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

      5. distribute-rgt-neg-in 89.3%

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

   9. Simplified 89.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.66 \cdot 10^{-102}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.02 \cdot 10^{-184}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{u}{\frac{t1}{v}} - v}{t1 + u}\\ \end{array} \]

Alternative 3: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-100} \lor \neg \left(t1 \leq 2.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5e-100) (not (<= t1 2.2e-121)))
   (/ (- v) (+ t1 u))
   (* (- v) (/ t1 (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e-100) || !(t1 <= 2.2e-121)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5d-100)) .or. (.not. (t1 <= 2.2d-121))) then
        tmp = -v / (t1 + u)
    else
        tmp = -v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e-100) || !(t1 <= 2.2e-121)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5e-100) or not (t1 <= 2.2e-121):
		tmp = -v / (t1 + u)
	else:
		tmp = -v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5e-100) || !(t1 <= 2.2e-121))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5e-100) || ~((t1 <= 2.2e-121)))
		tmp = -v / (t1 + u);
	else
		tmp = -v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5e-100], N[Not[LessEqual[t1, 2.2e-121]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.0000000000000001e-100 or 2.20000000000000021e-121 < t1

   1. Initial program 64.1%

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

      1. associate-/r* 82.6%

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

      2. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around inf 79.6%

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

      1. neg-mul-1 79.6%

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

   6. Simplified 79.6%

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

   if -5.0000000000000001e-100 < t1 < 2.20000000000000021e-121

   1. Initial program 79.9%

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

      1. associate-*l/ 83.4%

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

      2. *-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in t1 around 0 80.9%

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

      1. associate-*r/ 80.9%

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

      2. neg-mul-1 80.9%

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

      3. unpow2 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-100} \lor \neg \left(t1 \leq 2.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 4: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.3 \cdot 10^{-102} \lor \neg \left(t1 \leq 2.25 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.3e-102) (not (<= t1 2.25e-121)))
   (/ (- v) (+ t1 u))
   (* v (/ (/ (- t1) u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.3e-102) || !(t1 <= 2.25e-121)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.3d-102)) .or. (.not. (t1 <= 2.25d-121))) then
        tmp = -v / (t1 + u)
    else
        tmp = v * ((-t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.3e-102) || !(t1 <= 2.25e-121)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.3e-102) or not (t1 <= 2.25e-121):
		tmp = -v / (t1 + u)
	else:
		tmp = v * ((-t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.3e-102) || !(t1 <= 2.25e-121))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.3e-102) || ~((t1 <= 2.25e-121)))
		tmp = -v / (t1 + u);
	else
		tmp = v * ((-t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.3e-102], N[Not[LessEqual[t1, 2.25e-121]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.2999999999999997e-102 or 2.2500000000000002e-121 < t1

   1. Initial program 64.1%

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

      1. associate-/r* 82.6%

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

      2. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around inf 79.6%

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

      1. neg-mul-1 79.6%

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

   6. Simplified 79.6%

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

   if -4.2999999999999997e-102 < t1 < 2.2500000000000002e-121

   1. Initial program 79.9%

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

      1. associate-/l* 79.1%

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

      2. neg-mul-1 79.1%

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

      3. associate-*r/ 87.3%

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

      4. times-frac 91.8%

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

      5. div-inv 91.8%

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

      6. clear-num 91.8%

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

   3. Applied egg-rr 91.8%

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

   4. Taylor expanded in t1 around 0 77.4%

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

      1. *-commutative 77.4%

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

      2. unpow2 77.4%

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

      3. times-frac 89.0%

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

      4. *-commutative 89.0%

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

      5. associate-*l/ 87.1%

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

      6. associate-*r/ 88.4%

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

      7. associate-*r* 88.4%

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

      8. *-commutative 88.4%

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

      9. associate-*r/ 88.4%

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

      10. neg-mul-1 88.4%

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

      11. distribute-neg-frac 88.4%

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

   6. Simplified 88.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.3 \cdot 10^{-102} \lor \neg \left(t1 \leq 2.25 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \]

Alternative 5: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.85 \cdot 10^{-100} \lor \neg \left(t1 \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.85e-100) (not (<= t1 2.3e-68)))
   (/ (- v) (+ t1 u))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.85e-100) || !(t1 <= 2.3e-68)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.85d-100)) .or. (.not. (t1 <= 2.3d-68))) then
        tmp = -v / (t1 + u)
    else
        tmp = (t1 / u) * (-v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.85e-100) || !(t1 <= 2.3e-68)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.85e-100) or not (t1 <= 2.3e-68):
		tmp = -v / (t1 + u)
	else:
		tmp = (t1 / u) * (-v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.85e-100) || !(t1 <= 2.3e-68))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.85e-100) || ~((t1 <= 2.3e-68)))
		tmp = -v / (t1 + u);
	else
		tmp = (t1 / u) * (-v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.85e-100], N[Not[LessEqual[t1, 2.3e-68]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.84999999999999987e-100 or 2.29999999999999997e-68 < t1

   1. Initial program 62.5%

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

      1. associate-/r* 81.2%

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

      2. associate-/l* 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in t1 around inf 81.2%

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

      1. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

   if -3.84999999999999987e-100 < t1 < 2.29999999999999997e-68

   1. Initial program 80.4%

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

      1. associate-/r* 87.4%

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

      2. associate-/l* 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in v around 0 87.4%

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

      1. mul-1-neg 87.4%

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

      2. associate-*l/ 95.1%

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

      3. distribute-rgt-neg-out 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in t1 around 0 74.6%

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

      1. unpow2 74.6%

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

      2. times-frac 86.5%

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

      3. *-commutative 86.5%

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

      4. neg-mul-1 86.5%

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

      5. distribute-rgt-neg-in 86.5%

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

   9. Simplified 86.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.85 \cdot 10^{-100} \lor \neg \left(t1 \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 6: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+178} \lor \neg \left(u \leq 1.4 \cdot 10^{+99}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.2e+178) (not (<= u 1.4e+99)))
   (* v (/ t1 (* u u)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+178) || !(u <= 1.4e+99)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.2d+178)) .or. (.not. (u <= 1.4d+99))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+178) || !(u <= 1.4e+99)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.2e+178) or not (u <= 1.4e+99):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.2e+178) || !(u <= 1.4e+99))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.2e+178) || ~((u <= 1.4e+99)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.2e+178], N[Not[LessEqual[u, 1.4e+99]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.19999999999999997e178 or 1.4e99 < u

   1. Initial program 73.0%

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

      1. associate-*l/ 73.2%

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

      2. *-commutative 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in t1 around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

      3. unpow2 72.4%

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

   6. Simplified 72.4%

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

      1. add-sqr-sqrt 33.2%

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

      2. sqrt-unprod 66.3%

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

      3. sqr-neg 66.3%

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

      4. sqrt-unprod 39.1%

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

      5. times-frac 39.1%

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

   8. Applied egg-rr 39.1%

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

      1. times-frac 39.1%

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

      2. rem-square-sqrt 68.3%

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

   10. Simplified 68.3%

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

   if -2.19999999999999997e178 < u < 1.4e99

   1. Initial program 68.8%

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

      1. associate-/r* 81.2%

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

      2. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in t1 around inf 67.3%

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

      1. neg-mul-1 67.3%

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

   6. Simplified 67.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+178} \lor \neg \left(u \leq 1.4 \cdot 10^{+99}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+177}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+89}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.4e+177)
   (* v (/ t1 (* u u)))
   (if (<= u 3e+89) (/ (- v) (+ t1 u)) (/ t1 (* u (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.4e+177) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 3e+89) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-7.4d+177)) then
        tmp = v * (t1 / (u * u))
    else if (u <= 3d+89) then
        tmp = -v / (t1 + u)
    else
        tmp = t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.4e+177) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 3e+89) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.4e+177:
		tmp = v * (t1 / (u * u))
	elif u <= 3e+89:
		tmp = -v / (t1 + u)
	else:
		tmp = t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.4e+177)
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	elseif (u <= 3e+89)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.4e+177)
		tmp = v * (t1 / (u * u));
	elseif (u <= 3e+89)
		tmp = -v / (t1 + u);
	else
		tmp = t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.4e+177], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3e+89], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.40000000000000028e177

   1. Initial program 74.9%

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

      1. associate-*l/ 75.1%

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

      2. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in t1 around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. neg-mul-1 75.1%

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

      3. unpow2 75.1%

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

   6. Simplified 75.1%

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

      1. add-sqr-sqrt 26.9%

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

      2. sqrt-unprod 74.6%

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

      3. sqr-neg 74.6%

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

      4. sqrt-unprod 48.2%

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

      5. times-frac 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. times-frac 48.2%

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

      2. rem-square-sqrt 75.1%

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

   10. Simplified 75.1%

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

   if -7.40000000000000028e177 < u < 3.00000000000000013e89

   1. Initial program 68.1%

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

      1. associate-/r* 80.8%

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

      2. associate-/l* 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in t1 around inf 67.2%

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

      1. neg-mul-1 67.2%

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

   6. Simplified 67.2%

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

   if 3.00000000000000013e89 < u

   1. Initial program 74.3%

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

      1. associate-*l/ 74.6%

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

      2. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in t1 around 0 68.1%

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

      1. associate-*r/ 68.1%

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

      2. neg-mul-1 68.1%

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

      3. unpow2 68.1%

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

   6. Simplified 68.1%

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

      1. add-sqr-sqrt 33.0%

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

      2. sqrt-unprod 60.2%

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

      3. sqr-neg 60.2%

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

      4. sqrt-unprod 35.0%

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

      5. times-frac 35.0%

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

   8. Applied egg-rr 35.0%

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

      1. times-frac 35.0%

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

      2. rem-square-sqrt 62.8%

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

   10. Simplified 62.8%

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

       1. *-commutative 62.8%

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

       2. associate-/r/ 67.2%

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

       3. associate-/l* 67.0%

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

       4. associate-/r/ 67.0%

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

   12. Applied egg-rr 67.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+177}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+89}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 8: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+177}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.4e+177)
   (* v (/ t1 (* u u)))
   (if (<= u 3.5e+88) (/ (- v) (+ t1 u)) (/ t1 (/ (* u u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.4e+177) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 3.5e+88) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 / ((u * u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-7.4d+177)) then
        tmp = v * (t1 / (u * u))
    else if (u <= 3.5d+88) then
        tmp = -v / (t1 + u)
    else
        tmp = t1 / ((u * u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.4e+177) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 3.5e+88) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 / ((u * u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.4e+177:
		tmp = v * (t1 / (u * u))
	elif u <= 3.5e+88:
		tmp = -v / (t1 + u)
	else:
		tmp = t1 / ((u * u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.4e+177)
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	elseif (u <= 3.5e+88)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 / Float64(Float64(u * u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.4e+177)
		tmp = v * (t1 / (u * u));
	elseif (u <= 3.5e+88)
		tmp = -v / (t1 + u);
	else
		tmp = t1 / ((u * u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.4e+177], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.5e+88], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 / N[(N[(u * u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.40000000000000028e177

   1. Initial program 74.9%

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

      1. associate-*l/ 75.1%

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

      2. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in t1 around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. neg-mul-1 75.1%

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

      3. unpow2 75.1%

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

   6. Simplified 75.1%

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

      1. add-sqr-sqrt 26.9%

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

      2. sqrt-unprod 74.6%

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

      3. sqr-neg 74.6%

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

      4. sqrt-unprod 48.2%

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

      5. times-frac 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. times-frac 48.2%

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

      2. rem-square-sqrt 75.1%

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

   10. Simplified 75.1%

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

   if -7.40000000000000028e177 < u < 3.4999999999999998e88

   1. Initial program 68.1%

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

      1. associate-/r* 80.8%

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

      2. associate-/l* 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in t1 around inf 67.2%

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

      1. neg-mul-1 67.2%

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

   6. Simplified 67.2%

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

   if 3.4999999999999998e88 < u

   1. Initial program 74.3%

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

      1. associate-*l/ 74.6%

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

      2. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in t1 around 0 68.1%

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

      1. associate-*r/ 68.1%

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

      2. neg-mul-1 68.1%

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

      3. unpow2 68.1%

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

   6. Simplified 68.1%

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

      1. add-sqr-sqrt 33.0%

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

      2. sqrt-unprod 60.2%

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

      3. sqr-neg 60.2%

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

      4. sqrt-unprod 35.0%

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

      5. times-frac 35.0%

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

   8. Applied egg-rr 35.0%

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

      1. times-frac 35.0%

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

      2. rem-square-sqrt 62.8%

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

   10. Simplified 62.8%

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

       1. *-commutative 62.8%

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

       2. associate-/r/ 67.2%

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

       3. associate-/l* 67.0%

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

       4. associate-/r/ 67.0%

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

   12. Applied egg-rr 67.0%

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

       1. associate-*l/ 67.2%

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

   14. Applied egg-rr 67.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+177}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 9: 55.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3.5e+130) (/ (- v) t1) (/ (- v) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.5e+130) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.5d+130) then
        tmp = -v / t1
    else
        tmp = -v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.5e+130) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.5e+130:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.5e+130)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.5e+130)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.5e+130], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.5000000000000001e130

   1. Initial program 69.3%

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

      1. associate-*l/ 75.1%

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

      2. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in t1 around inf 59.5%

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

      1. associate-*r/ 59.5%

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

      2. neg-mul-1 59.5%

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

   6. Simplified 59.5%

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

   if 3.5000000000000001e130 < u

   1. Initial program 72.8%

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

      1. associate-/r* 90.6%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around 0 90.4%

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

      1. mul-1-neg 90.4%

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

      2. associate-/l* 94.0%

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

      3. distribute-neg-frac 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in t1 around inf 40.7%

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

      1. neg-mul-1 40.7%

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

      2. distribute-neg-frac 40.7%

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

   9. Simplified 40.7%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 10: 61.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. associate-/r* 83.8%

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

   2. associate-/l* 96.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in t1 around inf 60.1%

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

   1. neg-mul-1 60.1%

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

6. Simplified 60.1%

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

7. Final simplification 60.1%

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

Alternative 11: 53.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. associate-*l/ 74.5%

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

   2. *-commutative 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in t1 around inf 52.9%

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

   1. associate-*r/ 52.9%

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

   2. neg-mul-1 52.9%

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

6. Simplified 52.9%

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

7. Final simplification 52.9%

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

Alternative 12: 14.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. associate-*l/ 74.5%

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

   2. *-commutative 74.5%

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

3. Simplified 74.5%

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

   1. associate-/r* 95.8%

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

   2. associate-*r/ 97.9%

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

   3. *-commutative 97.9%

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

   4. associate-/r/ 96.1%

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

   5. div-inv 96.0%

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

   6. clear-num 96.0%

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

   7. associate-/r/ 96.0%

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

   8. clear-num 96.8%

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

   9. *-commutative 96.8%

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

   10. clear-num 96.0%

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

   11. div-inv 96.0%

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

   12. frac-2neg 96.0%

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

   13. remove-double-neg 96.0%

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

   14. associate-*l/ 97.7%

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

5. Applied egg-rr 56.4%

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

6. Taylor expanded in t1 around inf 11.5%

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

7. Final simplification 11.5%

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

Reproduce

herbie shell --seed 2023292 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))