Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.8%

Time: 11.4s

Alternatives: 15

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: 72.6% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{\frac{t1}{\left(-t1\right) - u} \cdot \sqrt{v\_m}}{\frac{t1 + u}{\sqrt{v\_m}}} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (* v_s (/ (* (/ t1 (- (- t1) u)) (sqrt v_m)) (/ (+ t1 u) (sqrt v_m)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (((t1 / (-t1 - u)) * sqrt(v_m)) / ((t1 + u) / sqrt(v_m)));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * (((t1 / (-t1 - u)) * sqrt(v_m)) / ((t1 + u) / sqrt(v_m)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * (((t1 / (-t1 - u)) * Math.sqrt(v_m)) / ((t1 + u) / Math.sqrt(v_m)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * (((t1 / (-t1 - u)) * math.sqrt(v_m)) / ((t1 + u) / math.sqrt(v_m)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(Float64(Float64(t1 / Float64(Float64(-t1) - u)) * sqrt(v_m)) / Float64(Float64(t1 + u) / sqrt(v_m))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * (((t1 / (-t1 - u)) * sqrt(v_m)) / ((t1 + u) / sqrt(v_m)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(N[(N[(t1 / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * N[Sqrt[v$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / N[Sqrt[v$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. times-frac 97.2%

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

   2. distribute-frac-neg 97.2%

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

   3. distribute-neg-frac2 97.2%

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

   4. +-commutative 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. unsub-neg 97.2%

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

3. Simplified 97.2%

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

   1. clear-num 96.6%

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

   2. associate-/r/ 97.0%

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

6. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l/ 97.2%

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

   3. *-un-lft-identity 97.2%

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

   4. clear-num 96.6%

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

   5. frac-2neg 96.6%

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

   6. frac-times 84.2%

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

   7. *-un-lft-identity 84.2%

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

   8. sub-neg 84.2%

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

   9. distribute-neg-in 84.2%

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

   10. +-commutative 84.2%

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

   11. remove-double-neg 84.2%

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

8. Applied egg-rr 84.2%

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

   1. neg-mul-1 84.2%

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

   2. add-sqr-sqrt 44.2%

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

   3. times-frac 44.2%

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

   4. associate-*l/ 39.4%

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

   5. sqrt-div 39.4%

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

   6. sqrt-unprod 21.6%

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

   7. add-sqr-sqrt 31.3%

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

   8. associate-*l/ 28.7%

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

   9. sqrt-div 29.1%

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

   10. sqrt-unprod 24.5%

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

   11. add-sqr-sqrt 52.9%

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

10. Applied egg-rr 52.9%

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

    1. associate-*l/ 52.9%

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

    2. associate-/r/ 52.6%

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

    3. associate-*r* 52.6%

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

    4. mul-1-neg 52.6%

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

12. Simplified 52.6%

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

13. Final simplification 52.6%

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

Alternative 2: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v\_m}{u}\\ t_2 := \frac{v\_m}{\left(-t1\right) - u \cdot 2}\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t1 \leq -58:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{-273}:\\ \;\;\;\;\frac{v\_m \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v_m u))) (t_2 (/ v_m (- (- t1) (* u 2.0)))))
   (*
    v_s
    (if (<= t1 -58.0)
      t_2
      (if (<= t1 -5e-254)
        t_1
        (if (<= t1 1.45e-273)
          (/ (* v_m (/ t1 u)) (- t1 u))
          (if (<= t1 9e-63) t_1 t_2)))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double t_1 = (-t1 / u) * (v_m / u);
	double t_2 = v_m / (-t1 - (u * 2.0));
	double tmp;
	if (t1 <= -58.0) {
		tmp = t_2;
	} else if (t1 <= -5e-254) {
		tmp = t_1;
	} else if (t1 <= 1.45e-273) {
		tmp = (v_m * (t1 / u)) / (t1 - u);
	} else if (t1 <= 9e-63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-t1 / u) * (v_m / u)
    t_2 = v_m / (-t1 - (u * 2.0d0))
    if (t1 <= (-58.0d0)) then
        tmp = t_2
    else if (t1 <= (-5d-254)) then
        tmp = t_1
    else if (t1 <= 1.45d-273) then
        tmp = (v_m * (t1 / u)) / (t1 - u)
    else if (t1 <= 9d-63) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double t_1 = (-t1 / u) * (v_m / u);
	double t_2 = v_m / (-t1 - (u * 2.0));
	double tmp;
	if (t1 <= -58.0) {
		tmp = t_2;
	} else if (t1 <= -5e-254) {
		tmp = t_1;
	} else if (t1 <= 1.45e-273) {
		tmp = (v_m * (t1 / u)) / (t1 - u);
	} else if (t1 <= 9e-63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	t_1 = (-t1 / u) * (v_m / u)
	t_2 = v_m / (-t1 - (u * 2.0))
	tmp = 0
	if t1 <= -58.0:
		tmp = t_2
	elif t1 <= -5e-254:
		tmp = t_1
	elif t1 <= 1.45e-273:
		tmp = (v_m * (t1 / u)) / (t1 - u)
	elif t1 <= 9e-63:
		tmp = t_1
	else:
		tmp = t_2
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v_m / u))
	t_2 = Float64(v_m / Float64(Float64(-t1) - Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -58.0)
		tmp = t_2;
	elseif (t1 <= -5e-254)
		tmp = t_1;
	elseif (t1 <= 1.45e-273)
		tmp = Float64(Float64(v_m * Float64(t1 / u)) / Float64(t1 - u));
	elseif (t1 <= 9e-63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	t_1 = (-t1 / u) * (v_m / u);
	t_2 = v_m / (-t1 - (u * 2.0));
	tmp = 0.0;
	if (t1 <= -58.0)
		tmp = t_2;
	elseif (t1 <= -5e-254)
		tmp = t_1;
	elseif (t1 <= 1.45e-273)
		tmp = (v_m * (t1 / u)) / (t1 - u);
	elseif (t1 <= 9e-63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v$95$m / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(v$95$s * If[LessEqual[t1, -58.0], t$95$2, If[LessEqual[t1, -5e-254], t$95$1, If[LessEqual[t1, 1.45e-273], N[(N[(v$95$m * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 9e-63], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -58 or 8.9999999999999999e-63 < t1

   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* 68.1%

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

      2. distribute-lft-neg-out 68.1%

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

      3. distribute-rgt-neg-in 68.1%

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

      4. associate-/r* 81.9%

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

      5. distribute-neg-frac2 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 95.5%

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

      9. *-un-lft-identity 95.5%

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

      10. frac-2neg 95.5%

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

      11. sub-neg 95.5%

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

      12. distribute-neg-in 95.5%

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

      13. +-commutative 95.5%

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

      14. remove-double-neg 95.5%

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

      15. add-sqr-sqrt 49.1%

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

      16. sqrt-unprod 40.0%

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

      17. sqr-neg 40.0%

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

      18. sqrt-unprod 19.6%

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

      19. add-sqr-sqrt 36.8%

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

      20. add-sqr-sqrt 15.1%

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

      21. sqrt-unprod 54.0%

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

   6. Applied egg-rr 95.5%

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

   7. Taylor expanded in u around 0 85.8%

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

      1. *-commutative 85.8%

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

   9. Simplified 85.8%

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

   if -58 < t1 < -5.0000000000000003e-254 or 1.44999999999999993e-273 < t1 < 8.9999999999999999e-63

   1. Initial program 81.4%

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

      1. times-frac 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-neg-frac2 97.5%

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

      4. +-commutative 97.5%

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

      5. distribute-neg-in 97.5%

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

      6. unsub-neg 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around 0 83.5%

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

      1. associate-*r/ 83.5%

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

      2. mul-1-neg 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in t1 around 0 84.5%

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

   if -5.0000000000000003e-254 < t1 < 1.44999999999999993e-273

   1. Initial program 85.2%

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

      1. associate-/l* 75.4%

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

      2. distribute-lft-neg-out 75.4%

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

      3. distribute-rgt-neg-in 75.4%

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

      4. associate-/r* 75.4%

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

      5. distribute-neg-frac2 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in t1 around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. clear-num 75.3%

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

      3. un-div-inv 77.2%

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

      4. add-sqr-sqrt 37.3%

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

      5. sqrt-unprod 63.8%

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

      6. sqr-neg 63.8%

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

      7. sqrt-unprod 26.5%

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

      8. add-sqr-sqrt 42.6%

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

   7. Applied egg-rr 42.6%

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

      1. frac-2neg 42.6%

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

      2. div-inv 42.6%

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

      3. associate-/r/ 42.7%

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

      4. distribute-rgt-neg-in 42.7%

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

      5. add-sqr-sqrt 21.1%

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

      6. sqrt-unprod 58.7%

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

      7. sqr-neg 58.7%

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

      8. sqrt-unprod 57.6%

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

      9. add-sqr-sqrt 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. add-sqr-sqrt 41.9%

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

      12. sqrt-unprod 99.5%

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

      13. sqr-neg 99.5%

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

      14. sqrt-unprod 57.6%

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

      15. add-sqr-sqrt 99.5%

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

      16. sub-neg 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

       3. *-commutative 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -58:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-254}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{-273}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -7.9 \cdot 10^{-163} \lor \neg \left(u \leq 1.35 \cdot 10^{-156}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v\_m}{\left(-t1\right) - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (or (<= u -7.9e-163) (not (<= u 1.35e-156)))
    (* t1 (/ (/ v_m (- (- t1) u)) (+ t1 u)))
    (/ v_m (- t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -7.9e-163) || !(u <= 1.35e-156)) {
		tmp = t1 * ((v_m / (-t1 - u)) / (t1 + u));
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.9d-163)) .or. (.not. (u <= 1.35d-156))) then
        tmp = t1 * ((v_m / (-t1 - u)) / (t1 + u))
    else
        tmp = v_m / -t1
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -7.9e-163) || !(u <= 1.35e-156)) {
		tmp = t1 * ((v_m / (-t1 - u)) / (t1 + u));
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if (u <= -7.9e-163) or not (u <= 1.35e-156):
		tmp = t1 * ((v_m / (-t1 - u)) / (t1 + u))
	else:
		tmp = v_m / -t1
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if ((u <= -7.9e-163) || !(u <= 1.35e-156))
		tmp = Float64(t1 * Float64(Float64(v_m / Float64(Float64(-t1) - u)) / Float64(t1 + u)));
	else
		tmp = Float64(v_m / Float64(-t1));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if ((u <= -7.9e-163) || ~((u <= 1.35e-156)))
		tmp = t1 * ((v_m / (-t1 - u)) / (t1 + u));
	else
		tmp = v_m / -t1;
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[Or[LessEqual[u, -7.9e-163], N[Not[LessEqual[u, 1.35e-156]], $MachinePrecision]], N[(t1 * N[(N[(v$95$m / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v$95$m / (-t1)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if u < -7.90000000000000049e-163 or 1.35000000000000006e-156 < u

   1. Initial program 78.6%

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

      1. associate-/l* 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

      4. associate-/r* 89.8%

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

      5. distribute-neg-frac2 89.8%

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

   3. Simplified 89.8%

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

   if -7.90000000000000049e-163 < u < 1.35000000000000006e-156

   1. Initial program 66.0%

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

      1. associate-/l* 61.8%

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

      2. distribute-lft-neg-out 61.8%

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

      3. distribute-rgt-neg-in 61.8%

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

      4. associate-/r* 73.9%

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

      5. distribute-neg-frac2 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in t1 around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. neg-mul-1 95.1%

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

   7. Simplified 95.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.9 \cdot 10^{-163} \lor \neg \left(u \leq 1.35 \cdot 10^{-156}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-t1\right) - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-103}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v\_m}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 13000000000000:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v\_m}{\left(-t1\right) - u}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (<= u -5.8e-103)
    (* (- t1) (/ (/ v_m u) (+ t1 u)))
    (if (<= u 13000000000000.0)
      (/ v_m (- t1))
      (* (/ t1 u) (/ v_m (- (- t1) u)))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (u <= -5.8e-103) {
		tmp = -t1 * ((v_m / u) / (t1 + u));
	} else if (u <= 13000000000000.0) {
		tmp = v_m / -t1;
	} else {
		tmp = (t1 / u) * (v_m / (-t1 - u));
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.8d-103)) then
        tmp = -t1 * ((v_m / u) / (t1 + u))
    else if (u <= 13000000000000.0d0) then
        tmp = v_m / -t1
    else
        tmp = (t1 / u) * (v_m / (-t1 - u))
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (u <= -5.8e-103) {
		tmp = -t1 * ((v_m / u) / (t1 + u));
	} else if (u <= 13000000000000.0) {
		tmp = v_m / -t1;
	} else {
		tmp = (t1 / u) * (v_m / (-t1 - u));
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if u <= -5.8e-103:
		tmp = -t1 * ((v_m / u) / (t1 + u))
	elif u <= 13000000000000.0:
		tmp = v_m / -t1
	else:
		tmp = (t1 / u) * (v_m / (-t1 - u))
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if (u <= -5.8e-103)
		tmp = Float64(Float64(-t1) * Float64(Float64(v_m / u) / Float64(t1 + u)));
	elseif (u <= 13000000000000.0)
		tmp = Float64(v_m / Float64(-t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v_m / Float64(Float64(-t1) - u)));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if (u <= -5.8e-103)
		tmp = -t1 * ((v_m / u) / (t1 + u));
	elseif (u <= 13000000000000.0)
		tmp = v_m / -t1;
	else
		tmp = (t1 / u) * (v_m / (-t1 - u));
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[LessEqual[u, -5.8e-103], N[((-t1) * N[(N[(v$95$m / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 13000000000000.0], N[(v$95$m / (-t1)), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v$95$m / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if u < -5.7999999999999997e-103

   1. Initial program 80.1%

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

      1. associate-/l* 83.9%

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

      2. distribute-lft-neg-out 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. associate-/r* 92.9%

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

      5. distribute-neg-frac2 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in t1 around 0 81.4%

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

   if -5.7999999999999997e-103 < u < 1.3e13

   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* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in t1 around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. neg-mul-1 84.4%

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

   7. Simplified 84.4%

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

   if 1.3e13 < u

   1. Initial program 77.5%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 86.4%

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

      1. associate-*r/ 86.4%

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

      2. mul-1-neg 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-103}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 13000000000000:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{t1}{t\_1 \cdot \frac{u}{v\_m}}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v\_m}{t\_1}\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)))
   (*
    v_s
    (if (<= u -5.8e-103)
      (/ t1 (* t_1 (/ u v_m)))
      (if (<= u 1.1e+14) (/ v_m (- t1)) (* (/ t1 u) (/ v_m t_1)))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (u <= -5.8e-103) {
		tmp = t1 / (t_1 * (u / v_m));
	} else if (u <= 1.1e+14) {
		tmp = v_m / -t1;
	} else {
		tmp = (t1 / u) * (v_m / t_1);
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 - u
    if (u <= (-5.8d-103)) then
        tmp = t1 / (t_1 * (u / v_m))
    else if (u <= 1.1d+14) then
        tmp = v_m / -t1
    else
        tmp = (t1 / u) * (v_m / t_1)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double t_1 = -t1 - u;
	double tmp;
	if (u <= -5.8e-103) {
		tmp = t1 / (t_1 * (u / v_m));
	} else if (u <= 1.1e+14) {
		tmp = v_m / -t1;
	} else {
		tmp = (t1 / u) * (v_m / t_1);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	t_1 = -t1 - u
	tmp = 0
	if u <= -5.8e-103:
		tmp = t1 / (t_1 * (u / v_m))
	elif u <= 1.1e+14:
		tmp = v_m / -t1
	else:
		tmp = (t1 / u) * (v_m / t_1)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	t_1 = Float64(Float64(-t1) - u)
	tmp = 0.0
	if (u <= -5.8e-103)
		tmp = Float64(t1 / Float64(t_1 * Float64(u / v_m)));
	elseif (u <= 1.1e+14)
		tmp = Float64(v_m / Float64(-t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v_m / t_1));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	t_1 = -t1 - u;
	tmp = 0.0;
	if (u <= -5.8e-103)
		tmp = t1 / (t_1 * (u / v_m));
	elseif (u <= 1.1e+14)
		tmp = v_m / -t1;
	else
		tmp = (t1 / u) * (v_m / t_1);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, N[(v$95$s * If[LessEqual[u, -5.8e-103], N[(t1 / N[(t$95$1 * N[(u / v$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.1e+14], N[(v$95$m / (-t1)), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v$95$m / t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if u < -5.7999999999999997e-103

   1. Initial program 80.1%

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

      1. times-frac 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l/ 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. clear-num 99.0%

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

      5. frac-2neg 99.0%

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

      6. frac-times 91.8%

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

      7. *-un-lft-identity 91.8%

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

      8. sub-neg 91.8%

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

      9. distribute-neg-in 91.8%

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

      10. +-commutative 91.8%

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

      11. remove-double-neg 91.8%

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

   8. Applied egg-rr 91.8%

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

   9. Taylor expanded in t1 around 0 81.5%

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

   if -5.7999999999999997e-103 < u < 1.1e14

   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* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in t1 around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. neg-mul-1 84.4%

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

   7. Simplified 84.4%

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

   if 1.1e14 < u

   1. Initial program 77.5%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 86.4%

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

      1. associate-*r/ 86.4%

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

      2. mul-1-neg 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{t1}{\left(\left(-t1\right) - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{-103}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v\_m}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v\_m}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (<= u -5.6e-103)
    (* (- t1) (/ (/ v_m u) (+ t1 u)))
    (if (<= u 3.2e+33) (/ v_m (- t1)) (/ (* t1 (/ v_m u)) (- t1 u))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (u <= -5.6e-103) {
		tmp = -t1 * ((v_m / u) / (t1 + u));
	} else if (u <= 3.2e+33) {
		tmp = v_m / -t1;
	} else {
		tmp = (t1 * (v_m / u)) / (t1 - u);
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.6d-103)) then
        tmp = -t1 * ((v_m / u) / (t1 + u))
    else if (u <= 3.2d+33) then
        tmp = v_m / -t1
    else
        tmp = (t1 * (v_m / u)) / (t1 - u)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (u <= -5.6e-103) {
		tmp = -t1 * ((v_m / u) / (t1 + u));
	} else if (u <= 3.2e+33) {
		tmp = v_m / -t1;
	} else {
		tmp = (t1 * (v_m / u)) / (t1 - u);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if u <= -5.6e-103:
		tmp = -t1 * ((v_m / u) / (t1 + u))
	elif u <= 3.2e+33:
		tmp = v_m / -t1
	else:
		tmp = (t1 * (v_m / u)) / (t1 - u)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if (u <= -5.6e-103)
		tmp = Float64(Float64(-t1) * Float64(Float64(v_m / u) / Float64(t1 + u)));
	elseif (u <= 3.2e+33)
		tmp = Float64(v_m / Float64(-t1));
	else
		tmp = Float64(Float64(t1 * Float64(v_m / u)) / Float64(t1 - u));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if (u <= -5.6e-103)
		tmp = -t1 * ((v_m / u) / (t1 + u));
	elseif (u <= 3.2e+33)
		tmp = v_m / -t1;
	else
		tmp = (t1 * (v_m / u)) / (t1 - u);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[LessEqual[u, -5.6e-103], N[((-t1) * N[(N[(v$95$m / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.2e+33], N[(v$95$m / (-t1)), $MachinePrecision], N[(N[(t1 * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if u < -5.60000000000000046e-103

   1. Initial program 80.1%

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

      1. associate-/l* 83.9%

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

      2. distribute-lft-neg-out 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. associate-/r* 92.9%

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

      5. distribute-neg-frac2 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in t1 around 0 81.4%

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

   if -5.60000000000000046e-103 < u < 3.20000000000000017e33

   1. Initial program 70.1%

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

      1. associate-/l* 66.7%

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

      2. distribute-lft-neg-out 66.7%

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

      3. distribute-rgt-neg-in 66.7%

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

      4. associate-/r* 78.4%

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

      5. distribute-neg-frac2 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in t1 around inf 84.1%

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

      1. associate-*r/ 84.1%

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

      2. neg-mul-1 84.1%

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

   7. Simplified 84.1%

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

   if 3.20000000000000017e33 < u

   1. Initial program 77.7%

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

      1. times-frac 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. distribute-neg-frac2 98.2%

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

      4. +-commutative 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. unsub-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around 0 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   7. Simplified 87.1%

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

      1. associate-*r/ 85.5%

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

      2. distribute-frac-neg 85.5%

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

      3. distribute-lft-neg-out 85.5%

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

      4. associate-/r/ 87.1%

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

      5. distribute-frac-neg2 87.1%

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

      6. frac-2neg 87.1%

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

      7. distribute-neg-frac 87.1%

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

      8. frac-2neg 87.1%

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

      9. div-inv 87.1%

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

      10. clear-num 87.1%

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

      11. distribute-neg-in 87.1%

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

      12. add-sqr-sqrt 41.9%

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

      13. sqrt-unprod 87.5%

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

      14. sqr-neg 87.5%

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

      15. sqrt-unprod 45.0%

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

      16. add-sqr-sqrt 87.1%

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

      17. sub-neg 87.1%

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

   9. Applied egg-rr 87.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{-103}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t1 \leq -60 \lor \neg \left(t1 \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{v\_m}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v\_m}{u}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (or (<= t1 -60.0) (not (<= t1 1.45e-10)))
    (/ v_m (- (- t1) u))
    (* (/ (- t1) u) (/ v_m u)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((t1 <= -60.0) || !(t1 <= 1.45e-10)) {
		tmp = v_m / (-t1 - u);
	} else {
		tmp = (-t1 / u) * (v_m / u);
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-60.0d0)) .or. (.not. (t1 <= 1.45d-10))) then
        tmp = v_m / (-t1 - u)
    else
        tmp = (-t1 / u) * (v_m / u)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((t1 <= -60.0) || !(t1 <= 1.45e-10)) {
		tmp = v_m / (-t1 - u);
	} else {
		tmp = (-t1 / u) * (v_m / u);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if (t1 <= -60.0) or not (t1 <= 1.45e-10):
		tmp = v_m / (-t1 - u)
	else:
		tmp = (-t1 / u) * (v_m / u)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if ((t1 <= -60.0) || !(t1 <= 1.45e-10))
		tmp = Float64(v_m / Float64(Float64(-t1) - u));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v_m / u));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if ((t1 <= -60.0) || ~((t1 <= 1.45e-10)))
		tmp = v_m / (-t1 - u);
	else
		tmp = (-t1 / u) * (v_m / u);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[Or[LessEqual[t1, -60.0], N[Not[LessEqual[t1, 1.45e-10]], $MachinePrecision]], N[(v$95$m / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t1 < -60 or 1.4499999999999999e-10 < t1

   1. Initial program 67.4%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 86.6%

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

   if -60 < t1 < 1.4499999999999999e-10

   1. Initial program 82.9%

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

      1. times-frac 94.3%

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

      2. distribute-frac-neg 94.3%

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

      3. distribute-neg-frac2 94.3%

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

      4. +-commutative 94.3%

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

      5. distribute-neg-in 94.3%

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

      6. unsub-neg 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in t1 around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. mul-1-neg 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in t1 around 0 82.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -60 \lor \neg \left(t1 \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t1 \leq -66 \lor \neg \left(t1 \leq 4.6 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v\_m}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v\_m}{u}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (or (<= t1 -66.0) (not (<= t1 4.6e-63)))
    (/ v_m (- (- t1) (* u 2.0)))
    (* (/ (- t1) u) (/ v_m u)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((t1 <= -66.0) || !(t1 <= 4.6e-63)) {
		tmp = v_m / (-t1 - (u * 2.0));
	} else {
		tmp = (-t1 / u) * (v_m / u);
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-66.0d0)) .or. (.not. (t1 <= 4.6d-63))) then
        tmp = v_m / (-t1 - (u * 2.0d0))
    else
        tmp = (-t1 / u) * (v_m / u)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((t1 <= -66.0) || !(t1 <= 4.6e-63)) {
		tmp = v_m / (-t1 - (u * 2.0));
	} else {
		tmp = (-t1 / u) * (v_m / u);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if (t1 <= -66.0) or not (t1 <= 4.6e-63):
		tmp = v_m / (-t1 - (u * 2.0))
	else:
		tmp = (-t1 / u) * (v_m / u)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if ((t1 <= -66.0) || !(t1 <= 4.6e-63))
		tmp = Float64(v_m / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v_m / u));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if ((t1 <= -66.0) || ~((t1 <= 4.6e-63)))
		tmp = v_m / (-t1 - (u * 2.0));
	else
		tmp = (-t1 / u) * (v_m / u);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[Or[LessEqual[t1, -66.0], N[Not[LessEqual[t1, 4.6e-63]], $MachinePrecision]], N[(v$95$m / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t1 < -66 or 4.6e-63 < t1

   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* 68.1%

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

      2. distribute-lft-neg-out 68.1%

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

      3. distribute-rgt-neg-in 68.1%

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

      4. associate-/r* 81.9%

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

      5. distribute-neg-frac2 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 95.5%

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

      9. *-un-lft-identity 95.5%

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

      10. frac-2neg 95.5%

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

      11. sub-neg 95.5%

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

      12. distribute-neg-in 95.5%

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

      13. +-commutative 95.5%

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

      14. remove-double-neg 95.5%

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

      15. add-sqr-sqrt 49.1%

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

      16. sqrt-unprod 40.0%

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

      17. sqr-neg 40.0%

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

      18. sqrt-unprod 19.6%

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

      19. add-sqr-sqrt 36.8%

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

      20. add-sqr-sqrt 15.1%

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

      21. sqrt-unprod 54.0%

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

   6. Applied egg-rr 95.5%

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

   7. Taylor expanded in u around 0 85.8%

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

      1. *-commutative 85.8%

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

   9. Simplified 85.8%

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

   if -66 < t1 < 4.6e-63

   1. Initial program 82.0%

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

      1. times-frac 93.8%

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

      2. distribute-frac-neg 93.8%

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

      3. distribute-neg-frac2 93.8%

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

      4. +-commutative 93.8%

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

      5. distribute-neg-in 93.8%

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

      6. unsub-neg 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t1 around 0 82.1%

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

      1. associate-*r/ 82.1%

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

      2. mul-1-neg 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in t1 around 0 83.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -66 \lor \neg \left(t1 \leq 4.6 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61} \lor \neg \left(u \leq 1.25 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{v\_m}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (or (<= u -2.8e+61) (not (<= u 1.25e+96)))
    (* (/ v_m u) -0.5)
    (/ v_m (- t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -2.8e+61) || !(u <= 1.25e+96)) {
		tmp = (v_m / u) * -0.5;
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.8d+61)) .or. (.not. (u <= 1.25d+96))) then
        tmp = (v_m / u) * (-0.5d0)
    else
        tmp = v_m / -t1
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -2.8e+61) || !(u <= 1.25e+96)) {
		tmp = (v_m / u) * -0.5;
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if (u <= -2.8e+61) or not (u <= 1.25e+96):
		tmp = (v_m / u) * -0.5
	else:
		tmp = v_m / -t1
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if ((u <= -2.8e+61) || !(u <= 1.25e+96))
		tmp = Float64(Float64(v_m / u) * -0.5);
	else
		tmp = Float64(v_m / Float64(-t1));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if ((u <= -2.8e+61) || ~((u <= 1.25e+96)))
		tmp = (v_m / u) * -0.5;
	else
		tmp = v_m / -t1;
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[Or[LessEqual[u, -2.8e+61], N[Not[LessEqual[u, 1.25e+96]], $MachinePrecision]], N[(N[(v$95$m / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v$95$m / (-t1)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if u < -2.8000000000000001e61 or 1.2500000000000001e96 < u

   1. Initial program 80.9%

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

      1. associate-/l* 80.1%

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

      2. distribute-lft-neg-out 80.1%

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

      3. distribute-rgt-neg-in 80.1%

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

      4. associate-/r* 93.4%

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

      5. distribute-neg-frac2 93.4%

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

   3. Simplified 93.4%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-*l/ 98.8%

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

      6. clear-num 98.8%

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

      7. frac-2neg 98.8%

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

      8. frac-times 85.4%

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

      9. *-un-lft-identity 85.4%

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

      10. frac-2neg 85.4%

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

      11. sub-neg 85.4%

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

      12. distribute-neg-in 85.4%

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

      13. +-commutative 85.4%

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

      14. remove-double-neg 85.4%

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

      15. add-sqr-sqrt 44.2%

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

      16. sqrt-unprod 72.3%

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

      17. sqr-neg 72.3%

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

      18. sqrt-unprod 36.8%

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

      19. add-sqr-sqrt 73.6%

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

      20. add-sqr-sqrt 35.6%

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

      21. sqrt-unprod 75.8%

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

   6. Applied egg-rr 85.4%

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

   7. Taylor expanded in u around 0 48.9%

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

      1. *-commutative 48.9%

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

   9. Simplified 48.9%

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

   10. Taylor expanded in t1 around 0 41.6%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -2.8000000000000001e61 < u < 1.2500000000000001e96

   1. Initial program 71.8%

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

      1. associate-/l* 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-/r* 80.8%

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

      5. distribute-neg-frac2 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t1 around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. neg-mul-1 72.9%

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

   7. Simplified 72.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61} \lor \neg \left(u \leq 1.25 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{v\_m}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 7 \cdot 10^{+95}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v\_m}}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (<= u -2.8e+61)
    (* (/ v_m u) -0.5)
    (if (<= u 7e+95) (/ v_m (- t1)) (/ 1.0 (/ u v_m))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (u <= -2.8e+61) {
		tmp = (v_m / u) * -0.5;
	} else if (u <= 7e+95) {
		tmp = v_m / -t1;
	} else {
		tmp = 1.0 / (u / v_m);
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.8d+61)) then
        tmp = (v_m / u) * (-0.5d0)
    else if (u <= 7d+95) then
        tmp = v_m / -t1
    else
        tmp = 1.0d0 / (u / v_m)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if (u <= -2.8e+61) {
		tmp = (v_m / u) * -0.5;
	} else if (u <= 7e+95) {
		tmp = v_m / -t1;
	} else {
		tmp = 1.0 / (u / v_m);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if u <= -2.8e+61:
		tmp = (v_m / u) * -0.5
	elif u <= 7e+95:
		tmp = v_m / -t1
	else:
		tmp = 1.0 / (u / v_m)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if (u <= -2.8e+61)
		tmp = Float64(Float64(v_m / u) * -0.5);
	elseif (u <= 7e+95)
		tmp = Float64(v_m / Float64(-t1));
	else
		tmp = Float64(1.0 / Float64(u / v_m));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if (u <= -2.8e+61)
		tmp = (v_m / u) * -0.5;
	elseif (u <= 7e+95)
		tmp = v_m / -t1;
	else
		tmp = 1.0 / (u / v_m);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[LessEqual[u, -2.8e+61], N[(N[(v$95$m / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 7e+95], N[(v$95$m / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if u < -2.8000000000000001e61

   1. Initial program 83.3%

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

      1. associate-/l* 83.5%

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

      2. distribute-lft-neg-out 83.5%

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

      3. distribute-rgt-neg-in 83.5%

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

      4. associate-/r* 95.5%

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

      5. distribute-neg-frac2 95.5%

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

   3. Simplified 95.5%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*l/ 99.8%

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

      6. clear-num 99.8%

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

      7. frac-2neg 99.8%

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

      8. frac-times 83.8%

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

      9. *-un-lft-identity 83.8%

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

      10. frac-2neg 83.8%

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

      11. sub-neg 83.8%

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

      12. distribute-neg-in 83.8%

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

      13. +-commutative 83.8%

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

      14. remove-double-neg 83.8%

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

      15. add-sqr-sqrt 48.4%

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

      16. sqrt-unprod 68.9%

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

      17. sqr-neg 68.9%

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

      18. sqrt-unprod 31.0%

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

      19. add-sqr-sqrt 73.1%

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

      20. add-sqr-sqrt 68.9%

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

      21. sqrt-unprod 73.3%

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

   6. Applied egg-rr 83.8%

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

   7. Taylor expanded in u around 0 50.2%

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

      1. *-commutative 50.2%

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

   9. Simplified 50.2%

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

   10. Taylor expanded in t1 around 0 40.2%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -2.8000000000000001e61 < u < 6.99999999999999999e95

   1. Initial program 71.8%

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

      1. associate-/l* 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-/r* 80.8%

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

      5. distribute-neg-frac2 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t1 around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. neg-mul-1 72.9%

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

   7. Simplified 72.9%

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

   if 6.99999999999999999e95 < u

   1. Initial program 78.3%

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

      1. associate-/l* 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

      4. associate-/r* 91.1%

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

      5. distribute-neg-frac2 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in t1 around 0 90.5%

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

   6. Taylor expanded in v around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. mul-1-neg 78.3%

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

      3. distribute-lft-neg-out 78.3%

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

      4. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in u around 0 51.8%

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

       1. clear-num 53.8%

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

       2. inv-pow 53.8%

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

       3. *-commutative 53.8%

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

       4. times-frac 44.9%

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

       5. add-sqr-sqrt 17.5%

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

       6. sqrt-unprod 53.2%

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

       7. sqr-neg 53.2%

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

       8. sqrt-unprod 27.1%

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

       9. add-sqr-sqrt 44.9%

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

   11. Applied egg-rr 44.9%

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

       1. unpow-1 44.9%

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

       2. *-inverses 44.9%

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

       3. *-rgt-identity 44.9%

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

   13. Simplified 44.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 7 \cdot 10^{+95}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61} \lor \neg \left(u \leq 1.2 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{v\_m}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (or (<= u -2.8e+61) (not (<= u 1.2e+96))) (/ v_m u) (/ v_m (- t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -2.8e+61) || !(u <= 1.2e+96)) {
		tmp = v_m / u;
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.8d+61)) .or. (.not. (u <= 1.2d+96))) then
        tmp = v_m / u
    else
        tmp = v_m / -t1
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -2.8e+61) || !(u <= 1.2e+96)) {
		tmp = v_m / u;
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if (u <= -2.8e+61) or not (u <= 1.2e+96):
		tmp = v_m / u
	else:
		tmp = v_m / -t1
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if ((u <= -2.8e+61) || !(u <= 1.2e+96))
		tmp = Float64(v_m / u);
	else
		tmp = Float64(v_m / Float64(-t1));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if ((u <= -2.8e+61) || ~((u <= 1.2e+96)))
		tmp = v_m / u;
	else
		tmp = v_m / -t1;
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[Or[LessEqual[u, -2.8e+61], N[Not[LessEqual[u, 1.2e+96]], $MachinePrecision]], N[(v$95$m / u), $MachinePrecision], N[(v$95$m / (-t1)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if u < -2.8000000000000001e61 or 1.19999999999999996e96 < u

   1. Initial program 80.9%

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

      1. associate-/l* 80.1%

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

      2. distribute-lft-neg-out 80.1%

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

      3. distribute-rgt-neg-in 80.1%

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

      4. associate-/r* 93.4%

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

      5. distribute-neg-frac2 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in t1 around 0 92.1%

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

      1. associate-*r/ 92.3%

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

      2. clear-num 92.2%

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

      3. un-div-inv 92.3%

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

      4. add-sqr-sqrt 45.1%

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

      5. sqrt-unprod 78.0%

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

      6. sqr-neg 78.0%

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

      7. sqrt-unprod 35.9%

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

      8. add-sqr-sqrt 70.3%

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

   7. Applied egg-rr 70.3%

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

   8. Taylor expanded in t1 around inf 41.3%

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

   if -2.8000000000000001e61 < u < 1.19999999999999996e96

   1. Initial program 71.8%

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

      1. associate-/l* 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-/r* 80.8%

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

      5. distribute-neg-frac2 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t1 around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. neg-mul-1 72.9%

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

   7. Simplified 72.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61} \lor \neg \left(u \leq 1.2 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61} \lor \neg \left(u \leq 1.02 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{v\_m}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v\_m}{-t1}\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (*
  v_s
  (if (or (<= u -2.8e+61) (not (<= u 1.02e+96)))
    (/ v_m (- u))
    (/ v_m (- t1)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -2.8e+61) || !(u <= 1.02e+96)) {
		tmp = v_m / -u;
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.8d+61)) .or. (.not. (u <= 1.02d+96))) then
        tmp = v_m / -u
    else
        tmp = v_m / -t1
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	double tmp;
	if ((u <= -2.8e+61) || !(u <= 1.02e+96)) {
		tmp = v_m / -u;
	} else {
		tmp = v_m / -t1;
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	tmp = 0
	if (u <= -2.8e+61) or not (u <= 1.02e+96):
		tmp = v_m / -u
	else:
		tmp = v_m / -t1
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	tmp = 0.0
	if ((u <= -2.8e+61) || !(u <= 1.02e+96))
		tmp = Float64(v_m / Float64(-u));
	else
		tmp = Float64(v_m / Float64(-t1));
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, u, v_m, t1)
	tmp = 0.0;
	if ((u <= -2.8e+61) || ~((u <= 1.02e+96)))
		tmp = v_m / -u;
	else
		tmp = v_m / -t1;
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * If[Or[LessEqual[u, -2.8e+61], N[Not[LessEqual[u, 1.02e+96]], $MachinePrecision]], N[(v$95$m / (-u)), $MachinePrecision], N[(v$95$m / (-t1)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if u < -2.8000000000000001e61 or 1.02000000000000001e96 < u

   1. Initial program 80.9%

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

      1. associate-/l* 80.1%

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

      2. distribute-lft-neg-out 80.1%

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

      3. distribute-rgt-neg-in 80.1%

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

      4. associate-/r* 93.4%

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

      5. distribute-neg-frac2 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in t1 around 0 92.1%

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

   6. Taylor expanded in t1 around inf 41.5%

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

      1. associate-*r/ 41.5%

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

      2. mul-1-neg 41.5%

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

   8. Simplified 41.5%

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

   if -2.8000000000000001e61 < u < 1.02000000000000001e96

   1. Initial program 71.8%

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

      1. associate-/l* 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-/r* 80.8%

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

      5. distribute-neg-frac2 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t1 around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. neg-mul-1 72.9%

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

   7. Simplified 72.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+61} \lor \neg \left(u \leq 1.02 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \left(\frac{t1}{t1 + u} \cdot \frac{v\_m}{\left(-t1\right) - u}\right) \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1)
 :precision binary64
 (* v_s (* (/ t1 (+ t1 u)) (/ v_m (- (- t1) u)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((t1 / (t1 + u)) * (v_m / (-t1 - u)));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * ((t1 / (t1 + u)) * (v_m / (-t1 - u)))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * ((t1 / (t1 + u)) * (v_m / (-t1 - u)));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * ((t1 / (t1 + u)) * (v_m / (-t1 - u)))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(Float64(t1 / Float64(t1 + u)) * Float64(v_m / Float64(Float64(-t1) - u))))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * ((t1 / (t1 + u)) * (v_m / (-t1 - u)));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v$95$m / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. times-frac 97.2%

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

   2. distribute-frac-neg 97.2%

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

   3. distribute-neg-frac2 97.2%

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

   4. +-commutative 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. unsub-neg 97.2%

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

3. Simplified 97.2%

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

5. Final simplification 97.2%

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

Alternative 14: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{v\_m}{\left(-t1\right) - u} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1) :precision binary64 (* v_s (/ v_m (- (- t1) u))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m / (-t1 - u));
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * (v_m / (-t1 - u))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m / (-t1 - u));
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * (v_m / (-t1 - u))

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(v_m / Float64(Float64(-t1) - u)))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * (v_m / (-t1 - u));
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(v$95$m / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. times-frac 97.2%

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

   2. distribute-frac-neg 97.2%

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

   3. distribute-neg-frac2 97.2%

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

   4. +-commutative 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. unsub-neg 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in t1 around inf 63.7%

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

6. Final simplification 63.7%

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

Alternative 15: 17.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \frac{v\_m}{u} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s u v_m t1) :precision binary64 (* v_s (/ v_m u)))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m / u);
}

Fortran

v\_m = abs(v)
v\_s = copysign(1.0d0, v)
real(8) function code(v_s, u, v_m, t1)
    real(8), intent (in) :: v_s
    real(8), intent (in) :: u
    real(8), intent (in) :: v_m
    real(8), intent (in) :: t1
    code = v_s * (v_m / u)
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double u, double v_m, double t1) {
	return v_s * (v_m / u);
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, u, v_m, t1):
	return v_s * (v_m / u)

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, u, v_m, t1)
	return Float64(v_s * Float64(v_m / u))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, u, v_m, t1)
	tmp = v_s * (v_m / u);
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, u_, v$95$m_, t1_] := N[(v$95$s * N[(v$95$m / u), $MachinePrecision]), $MachinePrecision]

Derivation

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* 74.4%

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

   2. distribute-lft-neg-out 74.4%

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

   3. distribute-rgt-neg-in 74.4%

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

   4. associate-/r* 85.1%

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

   5. distribute-neg-frac2 85.1%

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

3. Simplified 85.1%

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

5. Taylor expanded in t1 around 0 55.2%

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

   1. associate-*r/ 52.5%

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

   2. clear-num 52.7%

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

   3. un-div-inv 52.8%

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

   4. add-sqr-sqrt 27.7%

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

   5. sqrt-unprod 48.2%

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

   6. sqr-neg 48.2%

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

   7. sqrt-unprod 16.7%

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

   8. add-sqr-sqrt 31.1%

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

7. Applied egg-rr 31.1%

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

8. Taylor expanded in t1 around inf 18.2%

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

9. Final simplification 18.2%

   \[\leadsto \frac{v}{u} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))