Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A

Percentage Accurate: 99.1% → 99.1%

Time: 10.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

C

double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Python

def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

C

double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Python

def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

C

double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Python

def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Add Preprocessing

3. Final simplification 99.4%

   \[\leadsto 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
4. Add Preprocessing

Alternative 2: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-34}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.1e-60) 1.0 (if (<= y 2.75e-34) (+ 1.0 (/ (/ x t) (- y z))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.1e-60) {
		tmp = 1.0;
	} else if (y <= 2.75e-34) {
		tmp = 1.0 + ((x / t) / (y - z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.1d-60)) then
        tmp = 1.0d0
    else if (y <= 2.75d-34) then
        tmp = 1.0d0 + ((x / t) / (y - z))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.1e-60) {
		tmp = 1.0;
	} else if (y <= 2.75e-34) {
		tmp = 1.0 + ((x / t) / (y - z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.1e-60:
		tmp = 1.0
	elif y <= 2.75e-34:
		tmp = 1.0 + ((x / t) / (y - z))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.1e-60)
		tmp = 1.0;
	elseif (y <= 2.75e-34)
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.1e-60)
		tmp = 1.0;
	elseif (y <= 2.75e-34)
		tmp = 1.0 + ((x / t) / (y - z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.1e-60], 1.0, If[LessEqual[y, 2.75e-34], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.09999999999999988e-60 or 2.75000000000000007e-34 < y

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 93.5%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]

   4. Taylor expanded in y around 0 70.8%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot y}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{t \cdot y}\right)} \]

      2. associate-/r* 70.7%

         \[\leadsto 1 - \left(-\color{blue}{\frac{\frac{x}{t}}{y}}\right) \]

      3. distribute-neg-frac 70.7%

         \[\leadsto 1 - \color{blue}{\frac{-\frac{x}{t}}{y}} \]

      4. distribute-frac-neg 70.7%

         \[\leadsto 1 - \frac{\color{blue}{\frac{-x}{t}}}{y} \]

   6. Simplified 70.7%

      \[\leadsto 1 - \color{blue}{\frac{\frac{-x}{t}}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 69.4%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)\right)} \]

      2. expm1-udef 69.4%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)} - 1\right)} \]

      3. associate-/l/ 69.4%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{y \cdot t}}\right)} - 1\right) \]

      4. add-sqr-sqrt 30.2%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

      5. sqrt-unprod 59.3%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

      6. sqr-neg 59.3%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

      7. sqrt-unprod 39.4%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

      8. add-sqr-sqrt 68.6%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

   8. Applied egg-rr 68.6%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 68.6%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)\right)} \]

      2. expm1-log1p 69.3%

         \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   10. Simplified 69.3%

       \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   11. Taylor expanded in x around 0 89.0%

       \[\leadsto \color{blue}{1} \]

   if -3.09999999999999988e-60 < y < 2.75000000000000007e-34

   1. Initial program 98.6%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.6%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 98.6%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 98.6%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 96.5%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 96.5%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 96.5%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 96.5%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 96.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 96.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 96.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 96.5%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 96.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 96.5%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 96.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 96.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto 1 + \color{blue}{\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   7. Simplified 98.9%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   8. Taylor expanded in t around inf 88.0%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{x}{t \cdot \left(z - y\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 88.0%

         \[\leadsto 1 + \color{blue}{\left(-\frac{x}{t \cdot \left(z - y\right)}\right)} \]

      2. associate-/r* 88.4%

         \[\leadsto 1 + \left(-\color{blue}{\frac{\frac{x}{t}}{z - y}}\right) \]

      3. distribute-neg-frac 88.4%

         \[\leadsto 1 + \color{blue}{\frac{-\frac{x}{t}}{z - y}} \]

      4. distribute-frac-neg 88.4%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-x}{t}}}{z - y} \]

   10. Simplified 88.4%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-x}{t}}{z - y}} \]
   11. Step-by-step derivation

       1. frac-2neg 88.4%

          \[\leadsto 1 + \color{blue}{\frac{-\frac{-x}{t}}{-\left(z - y\right)}} \]

       2. div-inv 88.4%

          \[\leadsto 1 + \color{blue}{\left(-\frac{-x}{t}\right) \cdot \frac{1}{-\left(z - y\right)}} \]

       3. distribute-frac-neg 88.4%

          \[\leadsto 1 + \left(-\color{blue}{\left(-\frac{x}{t}\right)}\right) \cdot \frac{1}{-\left(z - y\right)} \]

       4. remove-double-neg 88.4%

          \[\leadsto 1 + \color{blue}{\frac{x}{t}} \cdot \frac{1}{-\left(z - y\right)} \]

       5. sub-neg 88.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{-\color{blue}{\left(z + \left(-y\right)\right)}} \]

       6. distribute-neg-in 88.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\color{blue}{\left(-z\right) + \left(-\left(-y\right)\right)}} \]

       7. add-sqr-sqrt 45.7%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)} \]

       8. sqrt-unprod 82.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} \]

       9. sqr-neg 82.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\sqrt{\color{blue}{y \cdot y}}\right)} \]

       10. sqrt-unprod 38.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)} \]

       11. add-sqr-sqrt 77.8%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{y}\right)} \]

       12. add-sqr-sqrt 39.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

       13. sqrt-unprod 80.5%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

       14. sqr-neg 80.5%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \sqrt{\color{blue}{y \cdot y}}} \]

       15. sqrt-unprod 42.8%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

       16. add-sqr-sqrt 88.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{y}} \]

   12. Applied egg-rr 88.4%

       \[\leadsto 1 + \color{blue}{\frac{x}{t} \cdot \frac{1}{\left(-z\right) + y}} \]
   13. Step-by-step derivation

       1. associate-*r/ 88.4%

          \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t} \cdot 1}{\left(-z\right) + y}} \]

       2. *-rgt-identity 88.4%

          \[\leadsto 1 + \frac{\color{blue}{\frac{x}{t}}}{\left(-z\right) + y} \]

       3. +-commutative 88.4%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y + \left(-z\right)}} \]

       4. unsub-neg 88.4%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y - z}} \]

   14. Simplified 88.4%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-34}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-294}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.28e-145)
   (+ 1.0 (/ (/ x (- y t)) z))
   (if (<= z 1.1e-294)
     (- 1.0 (/ x (* y (- y t))))
     (+ 1.0 (/ (/ x t) (- y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.28e-145) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else if (z <= 1.1e-294) {
		tmp = 1.0 - (x / (y * (y - t)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.28d-145)) then
        tmp = 1.0d0 + ((x / (y - t)) / z)
    else if (z <= 1.1d-294) then
        tmp = 1.0d0 - (x / (y * (y - t)))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.28e-145) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else if (z <= 1.1e-294) {
		tmp = 1.0 - (x / (y * (y - t)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.28e-145:
		tmp = 1.0 + ((x / (y - t)) / z)
	elif z <= 1.1e-294:
		tmp = 1.0 - (x / (y * (y - t)))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.28e-145)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - t)) / z));
	elseif (z <= 1.1e-294)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.28e-145)
		tmp = 1.0 + ((x / (y - t)) / z);
	elseif (z <= 1.1e-294)
		tmp = 1.0 - (x / (y * (y - t)));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.28e-145], N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-294], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.28e-145

   1. Initial program 98.3%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.3%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 98.3%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 98.3%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 98.9%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 98.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 98.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 98.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 87.8%

      \[\leadsto 1 + \color{blue}{\frac{x}{z \cdot \left(y - t\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto 1 + \frac{x}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. associate-/r* 88.2%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   7. Simplified 88.2%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   if -1.28e-145 < z < 1.1e-294

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.4%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]

   if 1.1e-294 < z

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 99.9%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 97.8%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 97.8%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 97.8%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 97.8%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 97.8%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 97.8%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 97.8%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto 1 + \color{blue}{\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 99.9%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   7. Simplified 99.9%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   8. Taylor expanded in t around inf 85.7%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{x}{t \cdot \left(z - y\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 85.7%

         \[\leadsto 1 + \color{blue}{\left(-\frac{x}{t \cdot \left(z - y\right)}\right)} \]

      2. associate-/r* 85.6%

         \[\leadsto 1 + \left(-\color{blue}{\frac{\frac{x}{t}}{z - y}}\right) \]

      3. distribute-neg-frac 85.6%

         \[\leadsto 1 + \color{blue}{\frac{-\frac{x}{t}}{z - y}} \]

      4. distribute-frac-neg 85.6%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-x}{t}}}{z - y} \]

   10. Simplified 85.6%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-x}{t}}{z - y}} \]
   11. Step-by-step derivation

       1. frac-2neg 85.6%

          \[\leadsto 1 + \color{blue}{\frac{-\frac{-x}{t}}{-\left(z - y\right)}} \]

       2. div-inv 85.6%

          \[\leadsto 1 + \color{blue}{\left(-\frac{-x}{t}\right) \cdot \frac{1}{-\left(z - y\right)}} \]

       3. distribute-frac-neg 85.6%

          \[\leadsto 1 + \left(-\color{blue}{\left(-\frac{x}{t}\right)}\right) \cdot \frac{1}{-\left(z - y\right)} \]

       4. remove-double-neg 85.6%

          \[\leadsto 1 + \color{blue}{\frac{x}{t}} \cdot \frac{1}{-\left(z - y\right)} \]

       5. sub-neg 85.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{-\color{blue}{\left(z + \left(-y\right)\right)}} \]

       6. distribute-neg-in 85.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\color{blue}{\left(-z\right) + \left(-\left(-y\right)\right)}} \]

       7. add-sqr-sqrt 36.3%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)} \]

       8. sqrt-unprod 81.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} \]

       9. sqr-neg 81.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\sqrt{\color{blue}{y \cdot y}}\right)} \]

       10. sqrt-unprod 46.0%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)} \]

       11. add-sqr-sqrt 79.3%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{y}\right)} \]

       12. add-sqr-sqrt 33.3%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

       13. sqrt-unprod 81.9%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

       14. sqr-neg 81.9%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \sqrt{\color{blue}{y \cdot y}}} \]

       15. sqrt-unprod 49.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

       16. add-sqr-sqrt 85.6%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{y}} \]

   12. Applied egg-rr 85.6%

       \[\leadsto 1 + \color{blue}{\frac{x}{t} \cdot \frac{1}{\left(-z\right) + y}} \]
   13. Step-by-step derivation

       1. associate-*r/ 85.6%

          \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t} \cdot 1}{\left(-z\right) + y}} \]

       2. *-rgt-identity 85.6%

          \[\leadsto 1 + \frac{\color{blue}{\frac{x}{t}}}{\left(-z\right) + y} \]

       3. +-commutative 85.6%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y + \left(-z\right)}} \]

       4. unsub-neg 85.6%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y - z}} \]

   14. Simplified 85.6%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y - z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-294}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-295}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.28e-145)
   (+ 1.0 (/ (/ x (- y t)) z))
   (if (<= z 9.5e-295)
     (- 1.0 (/ (/ x y) (- y t)))
     (+ 1.0 (/ (/ x t) (- y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.28e-145) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else if (z <= 9.5e-295) {
		tmp = 1.0 - ((x / y) / (y - t));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.28d-145)) then
        tmp = 1.0d0 + ((x / (y - t)) / z)
    else if (z <= 9.5d-295) then
        tmp = 1.0d0 - ((x / y) / (y - t))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.28e-145) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else if (z <= 9.5e-295) {
		tmp = 1.0 - ((x / y) / (y - t));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.28e-145:
		tmp = 1.0 + ((x / (y - t)) / z)
	elif z <= 9.5e-295:
		tmp = 1.0 - ((x / y) / (y - t))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.28e-145)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - t)) / z));
	elseif (z <= 9.5e-295)
		tmp = Float64(1.0 - Float64(Float64(x / y) / Float64(y - t)));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.28e-145)
		tmp = 1.0 + ((x / (y - t)) / z);
	elseif (z <= 9.5e-295)
		tmp = 1.0 - ((x / y) / (y - t));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.28e-145], N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e-295], N[(1.0 - N[(N[(x / y), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.28e-145

   1. Initial program 98.3%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.3%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 98.3%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 98.3%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 98.9%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 98.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 98.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 98.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 87.8%

      \[\leadsto 1 + \color{blue}{\frac{x}{z \cdot \left(y - t\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto 1 + \frac{x}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. associate-/r* 88.2%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   7. Simplified 88.2%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   if -1.28e-145 < z < 9.5e-295

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.4%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.4%

         \[\leadsto 1 - \color{blue}{\frac{\frac{x}{y}}{y - t}} \]

   5. Simplified 99.4%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{y}}{y - t}} \]

   if 9.5e-295 < z

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 99.9%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 97.8%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 97.8%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 97.8%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 97.8%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 97.8%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 97.8%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 97.8%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 97.8%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto 1 + \color{blue}{\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 99.9%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   7. Simplified 99.9%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   8. Taylor expanded in t around inf 85.7%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{x}{t \cdot \left(z - y\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 85.7%

         \[\leadsto 1 + \color{blue}{\left(-\frac{x}{t \cdot \left(z - y\right)}\right)} \]

      2. associate-/r* 85.6%

         \[\leadsto 1 + \left(-\color{blue}{\frac{\frac{x}{t}}{z - y}}\right) \]

      3. distribute-neg-frac 85.6%

         \[\leadsto 1 + \color{blue}{\frac{-\frac{x}{t}}{z - y}} \]

      4. distribute-frac-neg 85.6%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-x}{t}}}{z - y} \]

   10. Simplified 85.6%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-x}{t}}{z - y}} \]
   11. Step-by-step derivation

       1. frac-2neg 85.6%

          \[\leadsto 1 + \color{blue}{\frac{-\frac{-x}{t}}{-\left(z - y\right)}} \]

       2. div-inv 85.6%

          \[\leadsto 1 + \color{blue}{\left(-\frac{-x}{t}\right) \cdot \frac{1}{-\left(z - y\right)}} \]

       3. distribute-frac-neg 85.6%

          \[\leadsto 1 + \left(-\color{blue}{\left(-\frac{x}{t}\right)}\right) \cdot \frac{1}{-\left(z - y\right)} \]

       4. remove-double-neg 85.6%

          \[\leadsto 1 + \color{blue}{\frac{x}{t}} \cdot \frac{1}{-\left(z - y\right)} \]

       5. sub-neg 85.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{-\color{blue}{\left(z + \left(-y\right)\right)}} \]

       6. distribute-neg-in 85.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\color{blue}{\left(-z\right) + \left(-\left(-y\right)\right)}} \]

       7. add-sqr-sqrt 36.3%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)} \]

       8. sqrt-unprod 81.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} \]

       9. sqr-neg 81.6%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\sqrt{\color{blue}{y \cdot y}}\right)} \]

       10. sqrt-unprod 46.0%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)} \]

       11. add-sqr-sqrt 79.3%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{y}\right)} \]

       12. add-sqr-sqrt 33.3%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

       13. sqrt-unprod 81.9%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

       14. sqr-neg 81.9%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \sqrt{\color{blue}{y \cdot y}}} \]

       15. sqrt-unprod 49.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

       16. add-sqr-sqrt 85.6%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{y}} \]

   12. Applied egg-rr 85.6%

       \[\leadsto 1 + \color{blue}{\frac{x}{t} \cdot \frac{1}{\left(-z\right) + y}} \]
   13. Step-by-step derivation

       1. associate-*r/ 85.6%

          \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t} \cdot 1}{\left(-z\right) + y}} \]

       2. *-rgt-identity 85.6%

          \[\leadsto 1 + \frac{\color{blue}{\frac{x}{t}}}{\left(-z\right) + y} \]

       3. +-commutative 85.6%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y + \left(-z\right)}} \]

       4. unsub-neg 85.6%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y - z}} \]

   14. Simplified 85.6%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y - z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-295}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-143}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-294}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.7e-143)
   (+ 1.0 (/ (/ x (- y t)) z))
   (if (<= z 1.1e-294)
     (- 1.0 (/ (/ x y) (- y t)))
     (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-143) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else if (z <= 1.1e-294) {
		tmp = 1.0 - ((x / y) / (y - t));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.7d-143)) then
        tmp = 1.0d0 + ((x / (y - t)) / z)
    else if (z <= 1.1d-294) then
        tmp = 1.0d0 - ((x / y) / (y - t))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-143) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else if (z <= 1.1e-294) {
		tmp = 1.0 - ((x / y) / (y - t));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.7e-143:
		tmp = 1.0 + ((x / (y - t)) / z)
	elif z <= 1.1e-294:
		tmp = 1.0 - ((x / y) / (y - t))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.7e-143)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - t)) / z));
	elseif (z <= 1.1e-294)
		tmp = Float64(1.0 - Float64(Float64(x / y) / Float64(y - t)));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.7e-143)
		tmp = 1.0 + ((x / (y - t)) / z);
	elseif (z <= 1.1e-294)
		tmp = 1.0 - ((x / y) / (y - t));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.7e-143], N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-294], N[(1.0 - N[(N[(x / y), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999992e-143

   1. Initial program 98.3%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.3%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 98.3%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 98.3%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 98.9%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 98.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 98.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 98.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 98.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 87.8%

      \[\leadsto 1 + \color{blue}{\frac{x}{z \cdot \left(y - t\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto 1 + \frac{x}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. associate-/r* 88.2%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   7. Simplified 88.2%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   if -1.69999999999999992e-143 < z < 1.1e-294

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.4%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.4%

         \[\leadsto 1 - \color{blue}{\frac{\frac{x}{y}}{y - t}} \]

   5. Simplified 99.4%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{y}}{y - t}} \]

   if 1.1e-294 < z

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 85.7%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot \left(y - z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 85.7%

         \[\leadsto 1 - \color{blue}{\frac{-1 \cdot x}{t \cdot \left(y - z\right)}} \]

      2. neg-mul-1 85.7%

         \[\leadsto 1 - \frac{\color{blue}{-x}}{t \cdot \left(y - z\right)} \]

   5. Simplified 85.7%

      \[\leadsto 1 - \color{blue}{\frac{-x}{t \cdot \left(y - z\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-143}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-294}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.18 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-130}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.18e-89) 1.0 (if (<= t 1.55e-130) (+ 1.0 (/ x (* y z))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.18e-89) {
		tmp = 1.0;
	} else if (t <= 1.55e-130) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.18d-89)) then
        tmp = 1.0d0
    else if (t <= 1.55d-130) then
        tmp = 1.0d0 + (x / (y * z))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.18e-89) {
		tmp = 1.0;
	} else if (t <= 1.55e-130) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.18e-89:
		tmp = 1.0
	elif t <= 1.55e-130:
		tmp = 1.0 + (x / (y * z))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.18e-89)
		tmp = 1.0;
	elseif (t <= 1.55e-130)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.18e-89)
		tmp = 1.0;
	elseif (t <= 1.55e-130)
		tmp = 1.0 + (x / (y * z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.18e-89], 1.0, If[LessEqual[t, 1.55e-130], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if t < -1.18000000000000001e-89 or 1.55000000000000005e-130 < t

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 73.4%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]

   4. Taylor expanded in y around 0 66.5%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot y}} \]
   5. Step-by-step derivation

      1. mul-1-neg 66.5%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{t \cdot y}\right)} \]

      2. associate-/r* 66.5%

         \[\leadsto 1 - \left(-\color{blue}{\frac{\frac{x}{t}}{y}}\right) \]

      3. distribute-neg-frac 66.5%

         \[\leadsto 1 - \color{blue}{\frac{-\frac{x}{t}}{y}} \]

      4. distribute-frac-neg 66.5%

         \[\leadsto 1 - \frac{\color{blue}{\frac{-x}{t}}}{y} \]

   6. Simplified 66.5%

      \[\leadsto 1 - \color{blue}{\frac{\frac{-x}{t}}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 61.7%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)\right)} \]

      2. expm1-udef 61.7%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)} - 1\right)} \]

      3. associate-/l/ 61.7%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{y \cdot t}}\right)} - 1\right) \]

      4. add-sqr-sqrt 26.6%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

      5. sqrt-unprod 53.2%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

      6. sqr-neg 53.2%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

      7. sqrt-unprod 35.6%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

      8. add-sqr-sqrt 61.4%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

   8. Applied egg-rr 61.4%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 61.4%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)\right)} \]

      2. expm1-log1p 62.3%

         \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   10. Simplified 62.3%

       \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   11. Taylor expanded in x around 0 82.4%

       \[\leadsto \color{blue}{1} \]

   if -1.18000000000000001e-89 < t < 1.55000000000000005e-130

   1. Initial program 98.2%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.2%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 98.2%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 98.2%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 99.9%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 99.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 99.9%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 99.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 99.9%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 99.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 99.9%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 99.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 99.9%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 99.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 99.9%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 80.0%

      \[\leadsto 1 + \color{blue}{\frac{x}{z \cdot \left(y - t\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 80.0%

         \[\leadsto 1 + \frac{x}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. associate-/r* 81.8%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   7. Simplified 81.8%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   8. Taylor expanded in y around inf 74.6%

      \[\leadsto 1 + \color{blue}{\frac{x}{y \cdot z}} \]
   9. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto 1 + \frac{x}{\color{blue}{z \cdot y}} \]

   10. Simplified 74.6%

       \[\leadsto 1 + \color{blue}{\frac{x}{z \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.18 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-130}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-107}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-39}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.8e-107) 1.0 (if (<= y 5.4e-39) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.8e-107) {
		tmp = 1.0;
	} else if (y <= 5.4e-39) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9.8d-107)) then
        tmp = 1.0d0
    else if (y <= 5.4d-39) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.8e-107) {
		tmp = 1.0;
	} else if (y <= 5.4e-39) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9.8e-107:
		tmp = 1.0
	elif y <= 5.4e-39:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.8e-107)
		tmp = 1.0;
	elseif (y <= 5.4e-39)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9.8e-107)
		tmp = 1.0;
	elseif (y <= 5.4e-39)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.8e-107], 1.0, If[LessEqual[y, 5.4e-39], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.79999999999999959e-107 or 5.4000000000000001e-39 < y

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 92.5%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]

   4. Taylor expanded in y around 0 70.2%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot y}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.2%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{t \cdot y}\right)} \]

      2. associate-/r* 70.1%

         \[\leadsto 1 - \left(-\color{blue}{\frac{\frac{x}{t}}{y}}\right) \]

      3. distribute-neg-frac 70.1%

         \[\leadsto 1 - \color{blue}{\frac{-\frac{x}{t}}{y}} \]

      4. distribute-frac-neg 70.1%

         \[\leadsto 1 - \frac{\color{blue}{\frac{-x}{t}}}{y} \]

   6. Simplified 70.1%

      \[\leadsto 1 - \color{blue}{\frac{\frac{-x}{t}}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 67.5%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)\right)} \]

      2. expm1-udef 67.5%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)} - 1\right)} \]

      3. associate-/l/ 67.5%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{y \cdot t}}\right)} - 1\right) \]

      4. add-sqr-sqrt 29.5%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

      5. sqrt-unprod 57.9%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

      6. sqr-neg 57.9%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

      7. sqrt-unprod 38.3%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

      8. add-sqr-sqrt 66.8%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

   8. Applied egg-rr 66.8%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)\right)} \]

      2. expm1-log1p 67.5%

         \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   10. Simplified 67.5%

       \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   11. Taylor expanded in x around 0 87.6%

       \[\leadsto \color{blue}{1} \]

   if -9.79999999999999959e-107 < y < 5.4000000000000001e-39

   1. Initial program 98.5%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.1%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-107}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-39}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-106}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-37}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2e-106) 1.0 (if (<= y 1.85e-37) (- 1.0 (/ (/ x t) z)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-106) {
		tmp = 1.0;
	} else if (y <= 1.85e-37) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2d-106)) then
        tmp = 1.0d0
    else if (y <= 1.85d-37) then
        tmp = 1.0d0 - ((x / t) / z)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-106) {
		tmp = 1.0;
	} else if (y <= 1.85e-37) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2e-106:
		tmp = 1.0
	elif y <= 1.85e-37:
		tmp = 1.0 - ((x / t) / z)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2e-106)
		tmp = 1.0;
	elseif (y <= 1.85e-37)
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2e-106)
		tmp = 1.0;
	elseif (y <= 1.85e-37)
		tmp = 1.0 - ((x / t) / z);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2e-106], 1.0, If[LessEqual[y, 1.85e-37], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999988e-106 or 1.85e-37 < y

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 92.5%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]

   4. Taylor expanded in y around 0 70.2%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot y}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.2%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{t \cdot y}\right)} \]

      2. associate-/r* 70.1%

         \[\leadsto 1 - \left(-\color{blue}{\frac{\frac{x}{t}}{y}}\right) \]

      3. distribute-neg-frac 70.1%

         \[\leadsto 1 - \color{blue}{\frac{-\frac{x}{t}}{y}} \]

      4. distribute-frac-neg 70.1%

         \[\leadsto 1 - \frac{\color{blue}{\frac{-x}{t}}}{y} \]

   6. Simplified 70.1%

      \[\leadsto 1 - \color{blue}{\frac{\frac{-x}{t}}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 67.5%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)\right)} \]

      2. expm1-udef 67.5%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)} - 1\right)} \]

      3. associate-/l/ 67.5%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{y \cdot t}}\right)} - 1\right) \]

      4. add-sqr-sqrt 29.5%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

      5. sqrt-unprod 57.9%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

      6. sqr-neg 57.9%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

      7. sqrt-unprod 38.3%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

      8. add-sqr-sqrt 66.8%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

   8. Applied egg-rr 66.8%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 66.8%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)\right)} \]

      2. expm1-log1p 67.5%

         \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   10. Simplified 67.5%

       \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

   11. Taylor expanded in x around 0 87.6%

       \[\leadsto \color{blue}{1} \]

   if -1.99999999999999988e-106 < y < 1.85e-37

   1. Initial program 98.5%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.1%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 79.1%

         \[\leadsto 1 - \frac{\color{blue}{1 \cdot x}}{t \cdot z} \]

      2. times-frac 78.6%

         \[\leadsto 1 - \color{blue}{\frac{1}{t} \cdot \frac{x}{z}} \]

   5. Applied egg-rr 78.6%

      \[\leadsto 1 - \color{blue}{\frac{1}{t} \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

      1. associate-*l/ 78.6%

         \[\leadsto 1 - \color{blue}{\frac{1 \cdot \frac{x}{z}}{t}} \]

      2. *-lft-identity 78.6%

         \[\leadsto 1 - \frac{\color{blue}{\frac{x}{z}}}{t} \]

   7. Simplified 78.6%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{z}}{t}} \]

   8. Taylor expanded in x around 0 79.1%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
   9. Step-by-step derivation

      1. associate-/r* 79.5%

         \[\leadsto 1 - \color{blue}{\frac{\frac{x}{t}}{z}} \]

   10. Simplified 79.5%

       \[\leadsto 1 - \color{blue}{\frac{\frac{x}{t}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-106}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-37}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8.2e-50) (+ 1.0 (/ (/ x (- y t)) z)) (+ 1.0 (/ (/ x t) (- y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.2e-50) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 8.2d-50) then
        tmp = 1.0d0 + ((x / (y - t)) / z)
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.2e-50) {
		tmp = 1.0 + ((x / (y - t)) / z);
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 8.2e-50:
		tmp = 1.0 + ((x / (y - t)) / z)
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8.2e-50)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - t)) / z));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 8.2e-50)
		tmp = 1.0 + ((x / (y - t)) / z);
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8.2e-50], N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.19999999999999971e-50

   1. Initial program 99.2%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.2%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 99.2%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 99.2%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 99.5%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 99.5%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 99.5%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 99.5%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 99.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 99.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 99.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 99.5%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 99.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 99.5%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 99.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 99.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 80.3%

      \[\leadsto 1 + \color{blue}{\frac{x}{z \cdot \left(y - t\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 80.3%

         \[\leadsto 1 + \frac{x}{\color{blue}{\left(y - t\right) \cdot z}} \]

      2. associate-/r* 80.6%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   7. Simplified 80.6%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   if 8.19999999999999971e-50 < t

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      3. *-lft-identity 99.9%

         \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

      4. associate-/r* 95.5%

         \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

      5. associate-*r/ 95.5%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

      6. metadata-eval 95.5%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

      7. times-frac 95.5%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

      8. neg-mul-1 95.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      9. remove-double-neg 95.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

      10. neg-mul-1 95.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

      11. sub-neg 95.5%

          \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

      12. distribute-neg-out 95.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

      13. remove-double-neg 95.5%

          \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

      14. +-commutative 95.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

      15. sub-neg 95.5%

          \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

   3. Simplified 95.5%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto 1 + \color{blue}{\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   7. Simplified 99.8%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

   8. Taylor expanded in t around inf 98.4%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{x}{t \cdot \left(z - y\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 98.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{x}{t \cdot \left(z - y\right)}\right)} \]

      2. associate-/r* 98.3%

         \[\leadsto 1 + \left(-\color{blue}{\frac{\frac{x}{t}}{z - y}}\right) \]

      3. distribute-neg-frac 98.3%

         \[\leadsto 1 + \color{blue}{\frac{-\frac{x}{t}}{z - y}} \]

      4. distribute-frac-neg 98.3%

         \[\leadsto 1 + \frac{\color{blue}{\frac{-x}{t}}}{z - y} \]

   10. Simplified 98.3%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-x}{t}}{z - y}} \]
   11. Step-by-step derivation

       1. frac-2neg 98.3%

          \[\leadsto 1 + \color{blue}{\frac{-\frac{-x}{t}}{-\left(z - y\right)}} \]

       2. div-inv 98.4%

          \[\leadsto 1 + \color{blue}{\left(-\frac{-x}{t}\right) \cdot \frac{1}{-\left(z - y\right)}} \]

       3. distribute-frac-neg 98.4%

          \[\leadsto 1 + \left(-\color{blue}{\left(-\frac{x}{t}\right)}\right) \cdot \frac{1}{-\left(z - y\right)} \]

       4. remove-double-neg 98.4%

          \[\leadsto 1 + \color{blue}{\frac{x}{t}} \cdot \frac{1}{-\left(z - y\right)} \]

       5. sub-neg 98.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{-\color{blue}{\left(z + \left(-y\right)\right)}} \]

       6. distribute-neg-in 98.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\color{blue}{\left(-z\right) + \left(-\left(-y\right)\right)}} \]

       7. add-sqr-sqrt 50.7%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)} \]

       8. sqrt-unprod 94.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} \]

       9. sqr-neg 94.4%

          \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\sqrt{\color{blue}{y \cdot y}}\right)} \]

       10. sqrt-unprod 45.1%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)} \]

       11. add-sqr-sqrt 91.2%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \left(-\color{blue}{y}\right)} \]

       12. add-sqr-sqrt 46.2%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

       13. sqrt-unprod 92.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

       14. sqr-neg 92.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \sqrt{\color{blue}{y \cdot y}}} \]

       15. sqrt-unprod 47.7%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

       16. add-sqr-sqrt 98.4%

           \[\leadsto 1 + \frac{x}{t} \cdot \frac{1}{\left(-z\right) + \color{blue}{y}} \]

   12. Applied egg-rr 98.4%

       \[\leadsto 1 + \color{blue}{\frac{x}{t} \cdot \frac{1}{\left(-z\right) + y}} \]
   13. Step-by-step derivation

       1. associate-*r/ 98.3%

          \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t} \cdot 1}{\left(-z\right) + y}} \]

       2. *-rgt-identity 98.3%

          \[\leadsto 1 + \frac{\color{blue}{\frac{x}{t}}}{\left(-z\right) + y} \]

       3. +-commutative 98.3%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y + \left(-z\right)}} \]

       4. unsub-neg 98.3%

          \[\leadsto 1 + \frac{\frac{x}{t}}{\color{blue}{y - z}} \]

   14. Simplified 98.3%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{x}{y - t}}{z - y} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ 1.0 (/ (/ x (- y t)) (- z y))))

C

double code(double x, double y, double z, double t) {
	return 1.0 + ((x / (y - t)) / (z - y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + ((x / (y - t)) / (z - y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 + ((x / (y - t)) / (z - y));
}

Python

def code(x, y, z, t):
	return 1.0 + ((x / (y - t)) / (z - y))

Julia

function code(x, y, z, t)
	return Float64(1.0 + Float64(Float64(x / Float64(y - t)) / Float64(z - y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 + ((x / (y - t)) / (z - y));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Step-by-step derivation

   1. sub-neg 99.4%

      \[\leadsto \color{blue}{1 + \left(-\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)} \]

   2. distribute-frac-neg 99.4%

      \[\leadsto 1 + \color{blue}{\frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

   3. *-lft-identity 99.4%

      \[\leadsto 1 + \color{blue}{1 \cdot \frac{-x}{\left(y - z\right) \cdot \left(y - t\right)}} \]

   4. associate-/r* 98.5%

      \[\leadsto 1 + 1 \cdot \color{blue}{\frac{\frac{-x}{y - z}}{y - t}} \]

   5. associate-*r/ 98.5%

      \[\leadsto 1 + \color{blue}{\frac{1 \cdot \frac{-x}{y - z}}{y - t}} \]

   6. metadata-eval 98.5%

      \[\leadsto 1 + \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{-x}{y - z}}{y - t} \]

   7. times-frac 98.5%

      \[\leadsto 1 + \frac{\color{blue}{\frac{-1 \cdot \left(-x\right)}{-1 \cdot \left(y - z\right)}}}{y - t} \]

   8. neg-mul-1 98.5%

      \[\leadsto 1 + \frac{\frac{\color{blue}{-\left(-x\right)}}{-1 \cdot \left(y - z\right)}}{y - t} \]

   9. remove-double-neg 98.5%

      \[\leadsto 1 + \frac{\frac{\color{blue}{x}}{-1 \cdot \left(y - z\right)}}{y - t} \]

   10. neg-mul-1 98.5%

       \[\leadsto 1 + \frac{\frac{x}{\color{blue}{-\left(y - z\right)}}}{y - t} \]

   11. sub-neg 98.5%

       \[\leadsto 1 + \frac{\frac{x}{-\color{blue}{\left(y + \left(-z\right)\right)}}}{y - t} \]

   12. distribute-neg-out 98.5%

       \[\leadsto 1 + \frac{\frac{x}{\color{blue}{\left(-y\right) + \left(-\left(-z\right)\right)}}}{y - t} \]

   13. remove-double-neg 98.5%

       \[\leadsto 1 + \frac{\frac{x}{\left(-y\right) + \color{blue}{z}}}{y - t} \]

   14. +-commutative 98.5%

       \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z + \left(-y\right)}}}{y - t} \]

   15. sub-neg 98.5%

       \[\leadsto 1 + \frac{\frac{x}{\color{blue}{z - y}}}{y - t} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.4%

   \[\leadsto 1 + \color{blue}{\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 99.5%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

7. Simplified 99.5%

   \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]

8. Final simplification 99.5%

   \[\leadsto 1 + \frac{\frac{x}{y - t}}{z - y} \]
9. Add Preprocessing

Alternative 11: 75.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

def code(x, y, z, t):
	return 1.0

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 99.4%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 70.7%

   \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]

4. Taylor expanded in y around 0 57.0%

   \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot y}} \]
5. Step-by-step derivation

   1. mul-1-neg 57.0%

      \[\leadsto 1 - \color{blue}{\left(-\frac{x}{t \cdot y}\right)} \]

   2. associate-/r* 57.0%

      \[\leadsto 1 - \left(-\color{blue}{\frac{\frac{x}{t}}{y}}\right) \]

   3. distribute-neg-frac 57.0%

      \[\leadsto 1 - \color{blue}{\frac{-\frac{x}{t}}{y}} \]

   4. distribute-frac-neg 57.0%

      \[\leadsto 1 - \frac{\color{blue}{\frac{-x}{t}}}{y} \]

6. Simplified 57.0%

   \[\leadsto 1 - \color{blue}{\frac{\frac{-x}{t}}{y}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 51.0%

      \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)\right)} \]

   2. expm1-udef 51.0%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-x}{t}}{y}\right)} - 1\right)} \]

   3. associate-/l/ 51.0%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{y \cdot t}}\right)} - 1\right) \]

   4. add-sqr-sqrt 22.4%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

   5. sqrt-unprod 45.9%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

   6. sqr-neg 45.9%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

   7. sqrt-unprod 27.9%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

   8. add-sqr-sqrt 50.1%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

8. Applied egg-rr 50.1%

   \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 50.1%

      \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot t}\right)\right)} \]

   2. expm1-log1p 52.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

10. Simplified 52.2%

    \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot t}} \]

11. Taylor expanded in x around 0 74.0%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 74.0%

    \[\leadsto 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A"
  :precision binary64
  (- 1.0 (/ x (* (- y z) (- y t)))))