Rosa's DopplerBench

Percentage Accurate: 73.0% → 95.3%

Time: 12.7s

Alternatives: 14

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: 73.0% 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: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.8e+128)
   (/ (* t1 (/ v (+ t1 u))) (- t1 u))
   (* (- v) (/ (/ t1 (+ t1 u)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.8e+128) {
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	} else {
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6.8d+128)) then
        tmp = (t1 * (v / (t1 + u))) / (t1 - u)
    else
        tmp = -v * ((t1 / (t1 + u)) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.8e+128) {
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	} else {
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.8e+128:
		tmp = (t1 * (v / (t1 + u))) / (t1 - u)
	else:
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.8e+128)
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(t1 - u));
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / Float64(t1 + u)) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.8e+128)
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	else
		tmp = -v * ((t1 / (t1 + u)) / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.8e+128], N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.7999999999999997e128

   1. Initial program 78.7%

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

      1. associate-*l/ 77.2%

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

      2. *-commutative 77.2%

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

   3. Simplified 77.2%

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

      1. associate-/r* 85.6%

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

      2. associate-*r/ 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-/r/ 99.8%

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

      5. frac-2neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. remove-double-neg 99.8%

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

      8. div-inv 99.8%

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

      9. clear-num 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. add-sqr-sqrt 62.2%

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

      12. sqrt-unprod 93.5%

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

      13. sqr-neg 93.5%

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

      14. sqrt-unprod 37.7%

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

      15. add-sqr-sqrt 97.7%

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

      16. sub-neg 97.7%

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

   5. Applied egg-rr 97.7%

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

   if -6.7999999999999997e128 < u

   1. Initial program 70.7%

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

      1. associate-*l/ 77.6%

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

      2. *-commutative 77.6%

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

   3. Simplified 77.6%

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

      1. neg-mul-1 77.6%

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

      2. times-frac 97.5%

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

   5. Applied egg-rr 97.5%

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

      1. associate-*l/ 97.5%

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

      2. associate-*r/ 97.5%

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

      3. neg-mul-1 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \end{array} \]

Alternative 2: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -2.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{v \cdot \frac{u}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.22 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-166}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -2.4e+145)
     (/ (- (* v (/ u t1)) v) (+ t1 u))
     (if (<= t1 -1.22e-164)
       t_1
       (if (<= t1 4.4e-166)
         (* (/ v u) (/ t1 (- u)))
         (if (<= t1 1.6e+137) t_1 (/ (- v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -2.4e+145) {
		tmp = ((v * (u / t1)) - v) / (t1 + u);
	} else if (t1 <= -1.22e-164) {
		tmp = t_1;
	} else if (t1 <= 4.4e-166) {
		tmp = (v / u) * (t1 / -u);
	} else if (t1 <= 1.6e+137) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-2.4d+145)) then
        tmp = ((v * (u / t1)) - v) / (t1 + u)
    else if (t1 <= (-1.22d-164)) then
        tmp = t_1
    else if (t1 <= 4.4d-166) then
        tmp = (v / u) * (t1 / -u)
    else if (t1 <= 1.6d+137) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -2.4e+145) {
		tmp = ((v * (u / t1)) - v) / (t1 + u);
	} else if (t1 <= -1.22e-164) {
		tmp = t_1;
	} else if (t1 <= 4.4e-166) {
		tmp = (v / u) * (t1 / -u);
	} else if (t1 <= 1.6e+137) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -2.4e+145:
		tmp = ((v * (u / t1)) - v) / (t1 + u)
	elif t1 <= -1.22e-164:
		tmp = t_1
	elif t1 <= 4.4e-166:
		tmp = (v / u) * (t1 / -u)
	elif t1 <= 1.6e+137:
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -2.4e+145)
		tmp = Float64(Float64(Float64(v * Float64(u / t1)) - v) / Float64(t1 + u));
	elseif (t1 <= -1.22e-164)
		tmp = t_1;
	elseif (t1 <= 4.4e-166)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	elseif (t1 <= 1.6e+137)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -2.4e+145)
		tmp = ((v * (u / t1)) - v) / (t1 + u);
	elseif (t1 <= -1.22e-164)
		tmp = t_1;
	elseif (t1 <= 4.4e-166)
		tmp = (v / u) * (t1 / -u);
	elseif (t1 <= 1.6e+137)
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.4e+145], N[(N[(N[(v * N[(u / t1), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.22e-164], t$95$1, If[LessEqual[t1, 4.4e-166], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.6e+137], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.39999999999999992e145

   1. Initial program 37.4%

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

      1. associate-/r* 52.9%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 85.0%

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

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. associate-/l* 94.1%

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

   6. Simplified 94.1%

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

      1. associate-/r/ 94.1%

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

   8. Applied egg-rr 94.1%

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

   if -2.39999999999999992e145 < t1 < -1.2199999999999999e-164 or 4.4000000000000002e-166 < t1 < 1.60000000000000009e137

   1. Initial program 87.9%

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

      1. associate-*l/ 95.1%

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

      2. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   if -1.2199999999999999e-164 < t1 < 4.4000000000000002e-166

   1. Initial program 71.4%

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

      1. associate-/l* 75.7%

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

      2. neg-mul-1 75.7%

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

      3. associate-*r/ 87.7%

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

      4. times-frac 94.0%

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

      5. div-inv 93.9%

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

      6. clear-num 94.0%

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

   3. Applied egg-rr 94.0%

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

   4. Taylor expanded in t1 around 0 71.4%

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

      1. metadata-eval 71.4%

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

      2. *-commutative 71.4%

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

      3. unpow2 71.4%

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

      4. associate-/r* 80.7%

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

      5. *-commutative 80.7%

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

      6. associate-*l/ 81.0%

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

      7. times-frac 81.0%

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

      8. *-lft-identity 81.0%

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

      9. associate-*l/ 80.7%

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

      10. *-commutative 80.7%

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

      11. neg-mul-1 80.7%

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

      12. associate-/r* 71.4%

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

      13. times-frac 86.8%

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

   6. Simplified 86.8%

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

   if 1.60000000000000009e137 < t1

   1. Initial program 53.4%

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

      1. associate-/r* 61.7%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 90.2%

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

      1. neg-mul-1 90.2%

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

   6. Simplified 90.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{v \cdot \frac{u}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.22 \cdot 10^{-164}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-166}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+137}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-/l* 75.2%

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

   2. neg-mul-1 75.2%

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

   3. associate-*r/ 85.6%

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

   4. times-frac 97.1%

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

   5. div-inv 97.0%

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

   6. clear-num 97.2%

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

3. Applied egg-rr 97.2%

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

4. Final simplification 97.2%

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

Alternative 4: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.38 \cdot 10^{-14} \lor \neg \left(t1 \leq 6.9 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.38e-14) (not (<= t1 6.9e-40)))
   (/ -1.0 (/ (+ t1 u) v))
   (* v (/ (- t1) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.38e-14) || !(t1 <= 6.9e-40)) {
		tmp = -1.0 / ((t1 + u) / v);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.38d-14)) .or. (.not. (t1 <= 6.9d-40))) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else
        tmp = v * (-t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.38e-14) || !(t1 <= 6.9e-40)) {
		tmp = -1.0 / ((t1 + u) / v);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.38e-14) or not (t1 <= 6.9e-40):
		tmp = -1.0 / ((t1 + u) / v)
	else:
		tmp = v * (-t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.38e-14) || !(t1 <= 6.9e-40))
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.38e-14) || ~((t1 <= 6.9e-40)))
		tmp = -1.0 / ((t1 + u) / v);
	else
		tmp = v * (-t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.38e-14], N[Not[LessEqual[t1, 6.9e-40]], $MachinePrecision]], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.38000000000000002e-14 or 6.8999999999999996e-40 < t1

   1. Initial program 63.4%

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

      1. associate-/l* 69.4%

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

      2. neg-mul-1 69.4%

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

      3. associate-*r/ 82.4%

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

      4. times-frac 99.4%

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

      5. div-inv 99.3%

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

      6. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in t1 around inf 84.6%

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

      1. associate-*l/ 84.8%

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

      2. associate-/l* 85.0%

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

   6. Applied egg-rr 85.0%

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

   if -1.38000000000000002e-14 < t1 < 6.8999999999999996e-40

   1. Initial program 81.5%

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

      1. associate-*l/ 83.8%

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

      2. *-commutative 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in t1 around 0 75.3%

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

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

      3. unpow2 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.38 \cdot 10^{-14} \lor \neg \left(t1 \leq 6.9 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]

Alternative 5: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-14} \lor \neg \left(t1 \leq 3.3 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.45e-14) (not (<= t1 3.3e-39)))
   (/ -1.0 (/ (+ t1 u) v))
   (* v (/ (/ t1 u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e-14) || !(t1 <= 3.3e-39)) {
		tmp = -1.0 / ((t1 + u) / v);
	} else {
		tmp = v * ((t1 / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.45d-14)) .or. (.not. (t1 <= 3.3d-39))) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else
        tmp = v * ((t1 / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e-14) || !(t1 <= 3.3e-39)) {
		tmp = -1.0 / ((t1 + u) / v);
	} else {
		tmp = v * ((t1 / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.45e-14) or not (t1 <= 3.3e-39):
		tmp = -1.0 / ((t1 + u) / v)
	else:
		tmp = v * ((t1 / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.45e-14) || !(t1 <= 3.3e-39))
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.45e-14) || ~((t1 <= 3.3e-39)))
		tmp = -1.0 / ((t1 + u) / v);
	else
		tmp = v * ((t1 / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.45e-14], N[Not[LessEqual[t1, 3.3e-39]], $MachinePrecision]], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4500000000000001e-14 or 3.29999999999999985e-39 < t1

   1. Initial program 63.4%

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

      1. associate-/l* 69.4%

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

      2. neg-mul-1 69.4%

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

      3. associate-*r/ 82.4%

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

      4. times-frac 99.4%

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

      5. div-inv 99.3%

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

      6. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in t1 around inf 84.6%

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

      1. associate-*l/ 84.8%

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

      2. associate-/l* 85.0%

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

   6. Applied egg-rr 85.0%

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

   if -1.4500000000000001e-14 < t1 < 3.29999999999999985e-39

   1. Initial program 81.5%

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

      1. associate-*l/ 83.8%

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

      2. *-commutative 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in t1 around 0 75.3%

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

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

      3. unpow2 75.3%

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

   6. Simplified 75.3%

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

      1. neg-mul-1 75.3%

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

      2. times-frac 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. *-commutative 82.4%

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

      2. frac-2neg 82.4%

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

      3. metadata-eval 82.4%

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

      4. un-div-inv 82.5%

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-14} \lor \neg \left(t1 \leq 3.3 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 6: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.18 \cdot 10^{-14} \lor \neg \left(t1 \leq 7.6 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.18e-14) (not (<= t1 7.6e-41)))
   (/ -1.0 (/ (+ t1 u) v))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.18e-14) || !(t1 <= 7.6e-41)) {
		tmp = -1.0 / ((t1 + u) / v);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.18d-14)) .or. (.not. (t1 <= 7.6d-41))) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.18e-14) || !(t1 <= 7.6e-41)) {
		tmp = -1.0 / ((t1 + u) / v);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.18e-14) or not (t1 <= 7.6e-41):
		tmp = -1.0 / ((t1 + u) / v)
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.18e-14) || !(t1 <= 7.6e-41))
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.18e-14) || ~((t1 <= 7.6e-41)))
		tmp = -1.0 / ((t1 + u) / v);
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.18e-14], N[Not[LessEqual[t1, 7.6e-41]], $MachinePrecision]], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.17999999999999993e-14 or 7.59999999999999958e-41 < t1

   1. Initial program 63.4%

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

      1. associate-/l* 69.4%

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

      2. neg-mul-1 69.4%

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

      3. associate-*r/ 82.4%

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

      4. times-frac 99.4%

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

      5. div-inv 99.3%

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

      6. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in t1 around inf 84.6%

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

      1. associate-*l/ 84.8%

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

      2. associate-/l* 85.0%

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

   6. Applied egg-rr 85.0%

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

   if -1.17999999999999993e-14 < t1 < 7.59999999999999958e-41

   1. Initial program 81.5%

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

      1. associate-/l* 81.5%

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

      2. neg-mul-1 81.5%

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

      3. associate-*r/ 89.1%

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

      4. times-frac 94.7%

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

      5. div-inv 94.5%

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

      6. clear-num 94.6%

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

   3. Applied egg-rr 94.6%

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

   4. Taylor expanded in t1 around 0 73.8%

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

      1. metadata-eval 73.8%

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

      2. *-commutative 73.8%

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

      3. unpow2 73.8%

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

      4. associate-/r* 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*l/ 80.9%

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

      7. times-frac 80.9%

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

      8. *-lft-identity 80.9%

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

      9. associate-*l/ 80.1%

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

      10. *-commutative 80.1%

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

      11. neg-mul-1 80.1%

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

      12. associate-/r* 73.8%

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

      13. times-frac 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.18 \cdot 10^{-14} \lor \neg \left(t1 \leq 7.6 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-/r* 80.0%

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

   2. associate-/l* 97.2%

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

3. Simplified 97.2%

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

4. Final simplification 97.2%

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

Alternative 8: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+128} \lor \neg \left(u \leq 6.5 \cdot 10^{+201}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6e+128) (not (<= u 6.5e+201)))
   (* v (/ (/ t1 u) u))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6e+128) || !(u <= 6.5e+201)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6d+128)) .or. (.not. (u <= 6.5d+201))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6e+128) || !(u <= 6.5e+201)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6e+128) or not (u <= 6.5e+201):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6e+128) || !(u <= 6.5e+201))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6e+128) || ~((u <= 6.5e+201)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6e+128], N[Not[LessEqual[u, 6.5e+201]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.9999999999999997e128 or 6.5000000000000004e201 < u

   1. Initial program 83.0%

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

      1. associate-*l/ 82.1%

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

      2. *-commutative 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in t1 around 0 82.1%

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

      1. associate-*r/ 82.1%

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

      2. neg-mul-1 82.1%

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

      3. unpow2 82.1%

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

   6. Simplified 82.1%

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

      1. neg-mul-1 82.1%

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

      2. times-frac 83.7%

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

   8. Applied egg-rr 83.7%

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

      1. frac-times 82.1%

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

      2. neg-mul-1 82.1%

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

      3. add-sqr-sqrt 49.3%

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

      4. sqrt-unprod 68.5%

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

      5. sqr-neg 68.5%

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

      6. sqrt-unprod 32.8%

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

      7. add-sqr-sqrt 80.7%

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

      8. associate-/l/ 78.0%

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

   10. Applied egg-rr 78.0%

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

   if -5.9999999999999997e128 < u < 6.5000000000000004e201

   1. Initial program 68.2%

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

      1. associate-/r* 76.4%

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

      2. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around inf 66.0%

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

      1. neg-mul-1 66.0%

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

   6. Simplified 66.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+128} \lor \neg \left(u \leq 6.5 \cdot 10^{+201}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 9: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{t1 \cdot v}{u \cdot u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+201}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.2e+128)
   (/ (* t1 v) (* u u))
   (if (<= u 5.2e+201) (/ (- v) (+ t1 u)) (* v (/ (/ t1 u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.2e+128) {
		tmp = (t1 * v) / (u * u);
	} else if (u <= 5.2e+201) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6.2d+128)) then
        tmp = (t1 * v) / (u * u)
    else if (u <= 5.2d+201) then
        tmp = -v / (t1 + u)
    else
        tmp = v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.2e+128) {
		tmp = (t1 * v) / (u * u);
	} else if (u <= 5.2e+201) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.2e+128:
		tmp = (t1 * v) / (u * u)
	elif u <= 5.2e+201:
		tmp = -v / (t1 + u)
	else:
		tmp = v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.2e+128)
		tmp = Float64(Float64(t1 * v) / Float64(u * u));
	elseif (u <= 5.2e+201)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.2e+128)
		tmp = (t1 * v) / (u * u);
	elseif (u <= 5.2e+201)
		tmp = -v / (t1 + u);
	else
		tmp = v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.2e+128], N[(N[(t1 * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5.2e+201], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.20000000000000008e128

   1. Initial program 78.7%

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

      1. associate-*l/ 77.2%

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

      2. *-commutative 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in t1 around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. neg-mul-1 77.2%

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

      3. unpow2 77.2%

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

   6. Simplified 77.2%

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

      1. associate-*r/ 78.7%

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

      2. add-sqr-sqrt 47.3%

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

      3. sqrt-unprod 65.4%

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

      4. sqr-neg 65.4%

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

      5. sqrt-unprod 31.4%

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

      6. add-sqr-sqrt 74.4%

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

   8. Applied egg-rr 74.4%

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

   if -6.20000000000000008e128 < u < 5.19999999999999971e201

   1. Initial program 68.2%

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

      1. associate-/r* 76.4%

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

      2. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in t1 around inf 66.0%

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

      1. neg-mul-1 66.0%

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

   6. Simplified 66.0%

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

   if 5.19999999999999971e201 < u

   1. Initial program 91.3%

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

      1. associate-*l/ 91.8%

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

      2. *-commutative 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. neg-mul-1 91.8%

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

      3. unpow2 91.8%

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

   6. Simplified 91.8%

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

      1. neg-mul-1 91.8%

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

      2. times-frac 91.8%

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

   8. Applied egg-rr 91.8%

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

      1. frac-times 91.8%

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

      2. neg-mul-1 91.8%

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

      3. add-sqr-sqrt 56.5%

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

      4. sqrt-unprod 73.9%

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

      5. sqr-neg 73.9%

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

      6. sqrt-unprod 35.3%

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

      7. add-sqr-sqrt 91.8%

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

      8. associate-/l/ 91.8%

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

   10. Applied egg-rr 91.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{t1 \cdot v}{u \cdot u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+201}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 10: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+128} \lor \neg \left(u \leq 1.1 \cdot 10^{+178}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.2e+128) (not (<= u 1.1e+178))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+128) || !(u <= 1.1e+178)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.2d+128)) .or. (.not. (u <= 1.1d+178))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+128) || !(u <= 1.1e+178)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.2e+128) or not (u <= 1.1e+178):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.2e+128) || !(u <= 1.1e+178))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.2e+128) || ~((u <= 1.1e+178)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.2e+128], N[Not[LessEqual[u, 1.1e+178]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.20000000000000017e128 or 1.09999999999999999e178 < u

   1. Initial program 82.8%

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

      1. associate-/r* 89.2%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 87.6%

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

      1. mul-1-neg 87.6%

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

      2. associate-/l* 96.0%

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

      3. distribute-neg-frac 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in t1 around inf 49.0%

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

      1. neg-mul-1 49.0%

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

      2. distribute-neg-frac 49.0%

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

   9. Simplified 49.0%

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

   if -2.20000000000000017e128 < u < 1.09999999999999999e178

   1. Initial program 67.9%

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

      1. associate-*l/ 75.7%

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

      2. *-commutative 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in t1 around inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. neg-mul-1 64.0%

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

   6. Simplified 64.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+128} \lor \neg \left(u \leq 1.1 \cdot 10^{+178}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 62.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-/r* 80.0%

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

   2. associate-/l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in t1 around inf 62.1%

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

   1. neg-mul-1 62.1%

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

6. Simplified 62.1%

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

7. Final simplification 62.1%

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

Alternative 12: 62.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-/l* 75.2%

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

   2. neg-mul-1 75.2%

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

   3. associate-*r/ 85.6%

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

   4. times-frac 97.1%

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

   5. div-inv 97.0%

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

   6. clear-num 97.2%

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

3. Applied egg-rr 97.2%

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

4. Taylor expanded in t1 around inf 61.9%

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

   1. frac-2neg 61.9%

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

   2. metadata-eval 61.9%

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

   3. associate-*l/ 62.1%

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

   4. *-un-lft-identity 62.1%

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

   5. +-commutative 62.1%

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

   6. distribute-neg-in 62.1%

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

   7. add-sqr-sqrt 30.7%

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

   8. sqrt-unprod 69.1%

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

   9. sqr-neg 69.1%

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

   10. sqrt-unprod 31.3%

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

   11. add-sqr-sqrt 61.2%

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

6. Applied egg-rr 61.2%

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

   1. sub-neg 61.2%

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

8. Simplified 61.2%

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

9. Final simplification 61.2%

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

Alternative 13: 54.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-*l/ 77.5%

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

   2. *-commutative 77.5%

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

3. Simplified 77.5%

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

4. Taylor expanded in t1 around inf 51.3%

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

   1. associate-*r/ 51.3%

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

   2. neg-mul-1 51.3%

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

6. Simplified 51.3%

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

7. Final simplification 51.3%

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

Alternative 14: 13.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-/r* 80.0%

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

   2. associate-/l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in t1 around inf 47.2%

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

   1. +-commutative 47.2%

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

   2. neg-mul-1 47.2%

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

   3. unsub-neg 47.2%

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

   4. associate-/l* 48.6%

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

6. Simplified 48.6%

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

   1. associate-/r/ 49.3%

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

8. Applied egg-rr 49.3%

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

9. Taylor expanded in u around inf 15.1%

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

10. Final simplification 15.1%

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

Reproduce

herbie shell --seed 2023275 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))