Rosa's DopplerBench

Percentage Accurate: 73.3% → 97.9%

Time: 7.0s

Alternatives: 7

Speedup: 1.1×

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. *-commutative 70.0%

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

   2. times-frac 99.2%

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

   3. neg-mul-1 99.2%

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

   4. associate-/l* 99.2%

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

   5. associate-*r/ 99.2%

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

   6. associate-/l* 99.2%

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

   7. associate-/l/ 99.2%

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

   8. neg-mul-1 99.2%

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

   9. *-lft-identity 99.2%

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

   10. metadata-eval 99.2%

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

   11. times-frac 99.2%

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

   12. neg-mul-1 99.2%

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

   13. remove-double-neg 99.2%

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

   14. neg-mul-1 99.2%

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

   15. sub0-neg 99.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 99.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 99.2%

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

   18. div-sub 99.2%

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

   19. distribute-frac-neg 99.2%

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

   20. *-inverses 99.2%

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

   21. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-54} \lor \neg \left(t1 \leq 3.05 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e-54) (not (<= t1 3.05e-46)))
   (/ (- v) (+ t1 u))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-54) || !(t1 <= 3.05e-46)) {
		tmp = -v / (t1 + u);
	} 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 <= (-1d-54)) .or. (.not. (t1 <= 3.05d-46))) then
        tmp = -v / (t1 + u)
    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 <= -1e-54) || !(t1 <= 3.05e-46)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e-54) or not (t1 <= 3.05e-46):
		tmp = -v / (t1 + u)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e-54) || !(t1 <= 3.05e-46))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e-54) || ~((t1 <= 3.05e-46)))
		tmp = -v / (t1 + u);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e-54], N[Not[LessEqual[t1, 3.05e-46]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1e-54 or 3.05000000000000018e-46 < t1

   1. Initial program 65.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 85.7%

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

   if -1e-54 < t1 < 3.05000000000000018e-46

   1. Initial program 76.6%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 82.9%

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

      1. mul-1-neg 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in t1 around 0 85.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-54} \lor \neg \left(t1 \leq 3.05 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{-56} \lor \neg \left(t1 \leq 1.7 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.05e-56) (not (<= t1 1.7e-47)))
   (/ (- v) (+ t1 (* u 2.0)))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.05e-56) || !(t1 <= 1.7e-47)) {
		tmp = -v / (t1 + (u * 2.0));
	} 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 <= (-1.05d-56)) .or. (.not. (t1 <= 1.7d-47))) then
        tmp = -v / (t1 + (u * 2.0d0))
    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 <= -1.05e-56) || !(t1 <= 1.7e-47)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.05e-56) or not (t1 <= 1.7e-47):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.05e-56) || !(t1 <= 1.7e-47))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.05e-56) || ~((t1 <= 1.7e-47)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.05e-56], N[Not[LessEqual[t1, 1.7e-47]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.05000000000000003e-56 or 1.7000000000000001e-47 < t1

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 95.7%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in t1 around inf 86.5%

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

      1. *-commutative 86.5%

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

   9. Simplified 86.5%

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

   if -1.05000000000000003e-56 < t1 < 1.7000000000000001e-47

   1. Initial program 76.6%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 82.9%

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

      1. mul-1-neg 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in t1 around 0 85.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{-56} \lor \neg \left(t1 \leq 1.7 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-57} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.8e-57) (not (<= t1 1.35e-43)))
   (/ (- v) (+ t1 (* u 2.0)))
   (/ (/ v u) (/ (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.8e-57) || !(t1 <= 1.35e-43)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v / u) / (-u / 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 ((t1 <= (-3.8d-57)) .or. (.not. (t1 <= 1.35d-43))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (v / u) / (-u / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.8e-57) || !(t1 <= 1.35e-43)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v / u) / (-u / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.8e-57) or not (t1 <= 1.35e-43):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (v / u) / (-u / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.8e-57) || !(t1 <= 1.35e-43))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / u) / Float64(Float64(-u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.8e-57) || ~((t1 <= 1.35e-43)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (v / u) / (-u / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.8e-57], N[Not[LessEqual[t1, 1.35e-43]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] / N[((-u) / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.7999999999999997e-57 or 1.34999999999999996e-43 < t1

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 95.7%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in t1 around inf 86.5%

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

      1. *-commutative 86.5%

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

   9. Simplified 86.5%

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

   if -3.7999999999999997e-57 < t1 < 1.34999999999999996e-43

   1. Initial program 76.6%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 82.9%

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

      1. mul-1-neg 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in t1 around 0 85.3%

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

      1. neg-mul-1 85.3%

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

      2. clear-num 85.2%

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

      3. un-div-inv 85.2%

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

      4. times-frac 80.9%

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

      5. neg-mul-1 80.9%

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

      6. associate-/l/ 85.3%

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

      7. frac-2neg 85.3%

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

      8. frac-2neg 85.3%

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

      9. remove-double-neg 85.3%

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

      10. distribute-frac-neg 85.3%

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

      11. frac-2neg 85.3%

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

   9. Applied egg-rr 85.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-57} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \end{array} \]

Alternative 5: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+159}:\\ \;\;\;\;v \cdot 0\\ \mathbf{elif}\;u \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.7e+159)
   (* v 0.0)
   (if (<= u 1.3e+154) (/ (- v) (+ t1 u)) (* v 0.0))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e+159) {
		tmp = v * 0.0;
	} else if (u <= 1.3e+154) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.7d+159)) then
        tmp = v * 0.0d0
    else if (u <= 1.3d+154) then
        tmp = -v / (t1 + u)
    else
        tmp = v * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e+159) {
		tmp = v * 0.0;
	} else if (u <= 1.3e+154) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * 0.0;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.7e+159:
		tmp = v * 0.0
	elif u <= 1.3e+154:
		tmp = -v / (t1 + u)
	else:
		tmp = v * 0.0
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.7e+159)
		tmp = Float64(v * 0.0);
	elseif (u <= 1.3e+154)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.7e+159)
		tmp = v * 0.0;
	elseif (u <= 1.3e+154)
		tmp = -v / (t1 + u);
	else
		tmp = v * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.7e+159], N[(v * 0.0), $MachinePrecision], If[LessEqual[u, 1.3e+154], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.70000000000000008e159 or 1.29999999999999994e154 < u

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. times-frac 98.5%

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

      3. neg-mul-1 98.5%

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

      4. associate-/l* 98.5%

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

      5. associate-*r/ 98.5%

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

      6. associate-/l* 98.5%

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

      7. associate-/l/ 98.5%

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

      8. neg-mul-1 98.5%

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

      9. *-lft-identity 98.5%

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

      10. metadata-eval 98.5%

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

      11. times-frac 98.5%

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

      12. neg-mul-1 98.5%

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

      13. remove-double-neg 98.5%

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

      14. neg-mul-1 98.5%

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

      15. sub0-neg 98.5%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 98.5%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 98.5%

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

      18. div-sub 98.5%

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

      19. distribute-frac-neg 98.5%

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

      20. *-inverses 98.5%

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

      21. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in v around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. neg-mul-1 84.7%

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

   6. Simplified 84.7%

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

      1. distribute-rgt-in 84.7%

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

      2. *-un-lft-identity 84.7%

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

      3. associate-+l+ 84.7%

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

      4. clear-num 84.6%

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

      5. associate-*l/ 84.6%

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

      6. *-un-lft-identity 84.6%

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

   8. Applied egg-rr 84.6%

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

   9. Taylor expanded in u around 0 75.2%

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

       1. +-commutative 75.2%

          \[\leadsto \frac{-v}{t1 + \color{blue}{\left(2 \cdot u + \frac{{u}^{2}}{t1}\right)}} \]

       2. *-commutative 75.2%

          \[\leadsto \frac{-v}{t1 + \left(\color{blue}{u \cdot 2} + \frac{{u}^{2}}{t1}\right)} \]

       3. unpow2 75.2%

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

       4. associate-*r/ 84.7%

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

       5. distribute-lft-out 84.7%

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

   11. Simplified 84.7%

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

       1. expm1-log1p-u 84.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)\right)} \]

       2. expm1-udef 78.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1} \]

       3. add-sqr-sqrt 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       4. sqrt-unprod 70.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       5. sqr-neg 70.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{v \cdot v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       6. sqrt-unprod 36.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 75.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{v}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       8. +-commutative 75.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{v}{\color{blue}{u \cdot \left(2 + \frac{u}{t1}\right) + t1}}\right)} - 1 \]

       9. fma-def 75.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{v}{\color{blue}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}}\right)} - 1 \]

   13. Applied egg-rr 75.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\right)} - 1} \]

   15. Simplified 75.2%

       \[\leadsto \color{blue}{v \cdot 0} \]

   if -2.70000000000000008e159 < u < 1.29999999999999994e154

   1. Initial program 68.6%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 70.8%

      \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+159}:\\ \;\;\;\;v \cdot 0\\ \mathbf{elif}\;u \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot 0\\ \end{array} \]

Alternative 6: 68.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+103}:\\ \;\;\;\;v \cdot 0\\ \mathbf{elif}\;u \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8e+103) (* v 0.0) (if (<= u 3.9e+154) (/ (- v) t1) (* v 0.0))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8e+103) {
		tmp = v * 0.0;
	} else if (u <= 3.9e+154) {
		tmp = -v / t1;
	} else {
		tmp = v * 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-8d+103)) then
        tmp = v * 0.0d0
    else if (u <= 3.9d+154) then
        tmp = -v / t1
    else
        tmp = v * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8e+103) {
		tmp = v * 0.0;
	} else if (u <= 3.9e+154) {
		tmp = -v / t1;
	} else {
		tmp = v * 0.0;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8e+103:
		tmp = v * 0.0
	elif u <= 3.9e+154:
		tmp = -v / t1
	else:
		tmp = v * 0.0
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8e+103)
		tmp = Float64(v * 0.0);
	elseif (u <= 3.9e+154)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8e+103)
		tmp = v * 0.0;
	elseif (u <= 3.9e+154)
		tmp = -v / t1;
	else
		tmp = v * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8e+103], N[(v * 0.0), $MachinePrecision], If[LessEqual[u, 3.9e+154], N[((-v) / t1), $MachinePrecision], N[(v * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -8e103 or 3.9000000000000003e154 < u

   1. Initial program 75.1%

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

      1. *-commutative 75.1%

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

      2. times-frac 98.7%

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

      3. neg-mul-1 98.7%

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

      4. associate-/l* 98.7%

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

      5. associate-*r/ 98.7%

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

      6. associate-/l* 98.7%

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

      7. associate-/l/ 98.7%

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

      8. neg-mul-1 98.7%

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

      9. *-lft-identity 98.7%

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

      10. metadata-eval 98.7%

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

      11. times-frac 98.7%

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

      12. neg-mul-1 98.7%

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

      13. remove-double-neg 98.7%

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

      14. neg-mul-1 98.7%

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

      15. sub0-neg 98.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 98.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 98.7%

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

      18. div-sub 98.7%

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

      19. distribute-frac-neg 98.7%

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

      20. *-inverses 98.7%

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

      21. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in v around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. neg-mul-1 84.7%

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

   6. Simplified 84.7%

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

      1. distribute-rgt-in 84.7%

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

      2. *-un-lft-identity 84.7%

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

      3. associate-+l+ 84.7%

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

      4. clear-num 84.7%

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

      5. associate-*l/ 84.7%

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

      6. *-un-lft-identity 84.7%

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

   8. Applied egg-rr 84.7%

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

   9. Taylor expanded in u around 0 76.6%

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

       1. +-commutative 76.6%

          \[\leadsto \frac{-v}{t1 + \color{blue}{\left(2 \cdot u + \frac{{u}^{2}}{t1}\right)}} \]

       2. *-commutative 76.6%

          \[\leadsto \frac{-v}{t1 + \left(\color{blue}{u \cdot 2} + \frac{{u}^{2}}{t1}\right)} \]

       3. unpow2 76.6%

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

       4. associate-*r/ 84.7%

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

       5. distribute-lft-out 84.7%

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

   11. Simplified 84.7%

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

       1. expm1-log1p-u 84.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)\right)} \]

       2. expm1-udef 76.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1} \]

       3. add-sqr-sqrt 38.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       4. sqrt-unprod 69.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       5. sqr-neg 69.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{v \cdot v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       6. sqrt-unprod 34.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 72.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{v}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

       8. +-commutative 72.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{v}{\color{blue}{u \cdot \left(2 + \frac{u}{t1}\right) + t1}}\right)} - 1 \]

       9. fma-def 72.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{v}{\color{blue}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}}\right)} - 1 \]

   13. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\right)} - 1} \]

   15. Simplified 73.0%

       \[\leadsto \color{blue}{v \cdot 0} \]

   if -8e103 < u < 3.9000000000000003e154

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

      2. times-frac 99.4%

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

      3. neg-mul-1 99.4%

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

      4. associate-/l* 99.3%

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

      5. associate-*r/ 99.4%

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

      6. associate-/l* 99.4%

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

      7. associate-/l/ 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-lft-identity 99.4%

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

      10. metadata-eval 99.4%

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

      11. times-frac 99.4%

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

      12. neg-mul-1 99.4%

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

      13. remove-double-neg 99.4%

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

      14. neg-mul-1 99.4%

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

      15. sub0-neg 99.4%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.4%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.4%

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

      18. div-sub 99.4%

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

      19. distribute-frac-neg 99.4%

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

      20. *-inverses 99.4%

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

      21. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 69.7%

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

      1. mul-1-neg 69.7%

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

   6. Simplified 69.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+103}:\\ \;\;\;\;v \cdot 0\\ \mathbf{elif}\;u \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot 0\\ \end{array} \]

Alternative 7: 37.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ v \cdot 0 \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* v 0.0))

C

double code(double u, double v, double t1) {
	return v * 0.0;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v * 0.0d0
end function

Java

public static double code(double u, double v, double t1) {
	return v * 0.0;
}

Python

def code(u, v, t1):
	return v * 0.0

Julia

function code(u, v, t1)
	return Float64(v * 0.0)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v * 0.0;
end

Wolfram

code[u_, v_, t1_] := N[(v * 0.0), $MachinePrecision]

Derivation

1. Initial program 70.0%

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

   1. *-commutative 70.0%

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

   2. times-frac 99.2%

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

   3. neg-mul-1 99.2%

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

   4. associate-/l* 99.2%

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

   5. associate-*r/ 99.2%

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

   6. associate-/l* 99.2%

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

   7. associate-/l/ 99.2%

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

   8. neg-mul-1 99.2%

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

   9. *-lft-identity 99.2%

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

   10. metadata-eval 99.2%

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

   11. times-frac 99.2%

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

   12. neg-mul-1 99.2%

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

   13. remove-double-neg 99.2%

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

   14. neg-mul-1 99.2%

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

   15. sub0-neg 99.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 99.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 99.2%

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

   18. div-sub 99.2%

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

   19. distribute-frac-neg 99.2%

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

   20. *-inverses 99.2%

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

   21. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in v around 0 94.9%

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

   1. associate-*r/ 94.9%

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

   2. neg-mul-1 94.9%

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

6. Simplified 94.9%

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

   1. distribute-rgt-in 94.9%

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

   2. *-un-lft-identity 94.9%

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

   3. associate-+l+ 94.9%

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

   4. clear-num 94.9%

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

   5. associate-*l/ 94.9%

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

   6. *-un-lft-identity 94.9%

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

8. Applied egg-rr 94.9%

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

9. Taylor expanded in u around 0 88.7%

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

    1. +-commutative 88.7%

       \[\leadsto \frac{-v}{t1 + \color{blue}{\left(2 \cdot u + \frac{{u}^{2}}{t1}\right)}} \]

    2. *-commutative 88.7%

       \[\leadsto \frac{-v}{t1 + \left(\color{blue}{u \cdot 2} + \frac{{u}^{2}}{t1}\right)} \]

    3. unpow2 88.7%

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

    4. associate-*r/ 94.9%

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

    5. distribute-lft-out 94.9%

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

11. Simplified 94.9%

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

    1. expm1-log1p-u 79.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)\right)} \]

    2. expm1-udef 47.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1} \]

    3. add-sqr-sqrt 23.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

    4. sqrt-unprod 35.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

    5. sqr-neg 35.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{v \cdot v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

    6. sqrt-unprod 16.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

    7. add-sqr-sqrt 32.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{v}}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\right)} - 1 \]

    8. +-commutative 32.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{v}{\color{blue}{u \cdot \left(2 + \frac{u}{t1}\right) + t1}}\right)} - 1 \]

    9. fma-def 32.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{v}{\color{blue}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}}\right)} - 1 \]

13. Applied egg-rr 32.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v}{\mathsf{fma}\left(u, 2 + \frac{u}{t1}, t1\right)}\right)} - 1} \]

15. Simplified 33.3%

    \[\leadsto \color{blue}{v \cdot 0} \]

16. Final simplification 33.3%

    \[\leadsto v \cdot 0 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))