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

Percentage Accurate: 99.1% → 99.1%

Time: 10.3s

Alternatives: 12

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 99.7%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ t_2 := 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- t y))))
        (t_2 (- 1.0 (/ x (* (- y z) (- y t))))))
   (if (<= t_2 -1.0) t_1 (if (<= t_2 2.0) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (t - y));
	double t_2 = 1.0 - (x / ((y - z) * (y - t)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (t - y))
    t_2 = 1.0d0 - (x / ((y - z) * (y - t)))
    if (t_2 <= (-1.0d0)) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (t - y));
	double t_2 = 1.0 - (x / ((y - z) * (y - t)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (t - y))
	t_2 = 1.0 - (x / ((y - z) * (y - t)))
	tmp = 0
	if t_2 <= -1.0:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(t - y)))
	t_2 = Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (t - y));
	t_2 = 1.0 - (x / ((y - z) * (y - t)));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -1 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 93.6

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

   5. Simplified 93.6%

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

   if -1 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -1:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{elif}\;1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -20000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* z (- y t)))))
   (if (<= t_1 -20000000.0) t_2 (if (<= t_1 0.4) 1.0 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (z * (y - t));
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / (z * (y - t))
    if (t_1 <= (-20000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.4d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (z * (y - t));
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / (z * (y - t))
	tmp = 0
	if t_1 <= -20000000.0:
		tmp = t_2
	elif t_1 <= 0.4:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(z * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -20000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / (z * (y - t));
	tmp = 0.0;
	if (t_1 <= -20000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000.0], t$95$2, If[LessEqual[t$95$1, 0.4], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -2e7 or 0.40000000000000002 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 98.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-frac-neg2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 98.2

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 63.7

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

   7. Simplified 63.7%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 60.8

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

   10. Simplified 60.8%

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

   if -2e7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -20000000:\\ \;\;\;\;\frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := 1 - \frac{x}{z \cdot t}\\ \mathbf{if}\;t\_1 \leq -20000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (- 1.0 (/ x (* z t)))))
   (if (<= t_1 -20000000.0) t_2 (if (<= t_1 1.5e-8) 1.0 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = 1.0 - (x / (z * t));
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1.5e-8) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = 1.0d0 - (x / (z * t))
    if (t_1 <= (-20000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 1.5d-8) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = 1.0 - (x / (z * t));
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1.5e-8) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = 1.0 - (x / (z * t))
	tmp = 0
	if t_1 <= -20000000.0:
		tmp = t_2
	elif t_1 <= 1.5e-8:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(1.0 - Float64(x / Float64(z * t)))
	tmp = 0.0
	if (t_1 <= -20000000.0)
		tmp = t_2;
	elseif (t_1 <= 1.5e-8)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = 1.0 - (x / (z * t));
	tmp = 0.0;
	if (t_1 <= -20000000.0)
		tmp = t_2;
	elseif (t_1 <= 1.5e-8)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000.0], t$95$2, If[LessEqual[t$95$1, 1.5e-8], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -2e7 or 1.49999999999999987e-8 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 41.8

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

   5. Simplified 41.8%

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

   if -2e7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 1.49999999999999987e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -20000000:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -5e-5)
     (+ 1.0 (/ x (* y t)))
     (if (<= t_1 0.4) 1.0 (+ 1.0 (/ x (* y z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -5e-5) {
		tmp = 1.0 + (x / (y * t));
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-5d-5)) then
        tmp = 1.0d0 + (x / (y * t))
    else if (t_1 <= 0.4d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (x / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -5e-5) {
		tmp = 1.0 + (x / (y * t));
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -5e-5:
		tmp = 1.0 + (x / (y * t))
	elif t_1 <= 0.4:
		tmp = 1.0
	else:
		tmp = 1.0 + (x / (y * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -5e-5)
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -5e-5)
		tmp = 1.0 + (x / (y * t));
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = 1.0 + (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-5], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.4], 1.0, N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -5.00000000000000024e-5

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      9. *-lowering-*.f64 72.6

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \left(0 - t\right) - z, \color{blue}{t \cdot z}\right)} \]

   5. Simplified 72.6%

      \[\leadsto 1 - \frac{x}{\color{blue}{\mathsf{fma}\left(y, \left(0 - t\right) - z, t \cdot z\right)}} \]

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 24.6

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

   8. Simplified 24.6%

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

   if -5.00000000000000024e-5 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.7%

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

   if 0.40000000000000002 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 40.4

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

   8. Simplified 40.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := 1 + \frac{x}{y \cdot t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (+ 1.0 (/ x (* y t)))))
   (if (<= t_1 -5e-5) t_2 (if (<= t_1 0.4) 1.0 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = 1.0 + (x / (y * t));
	double tmp;
	if (t_1 <= -5e-5) {
		tmp = t_2;
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = 1.0d0 + (x / (y * t))
    if (t_1 <= (-5d-5)) then
        tmp = t_2
    else if (t_1 <= 0.4d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = 1.0 + (x / (y * t));
	double tmp;
	if (t_1 <= -5e-5) {
		tmp = t_2;
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = 1.0 + (x / (y * t))
	tmp = 0
	if t_1 <= -5e-5:
		tmp = t_2
	elif t_1 <= 0.4:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(1.0 + Float64(x / Float64(y * t)))
	tmp = 0.0
	if (t_1 <= -5e-5)
		tmp = t_2;
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = 1.0 + (x / (y * t));
	tmp = 0.0;
	if (t_1 <= -5e-5)
		tmp = t_2;
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-5], t$95$2, If[LessEqual[t$95$1, 0.4], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -5.00000000000000024e-5 or 0.40000000000000002 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      9. *-lowering-*.f64 73.0

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \left(0 - t\right) - z, \color{blue}{t \cdot z}\right)} \]

   5. Simplified 73.0%

      \[\leadsto 1 - \frac{x}{\color{blue}{\mathsf{fma}\left(y, \left(0 - t\right) - z, t \cdot z\right)}} \]

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 23.1

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

   8. Simplified 23.1%

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

   if -5.00000000000000024e-5 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -4e+26) (/ x (* y t)) (if (<= t_1 0.4) 1.0 (/ x (* y z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -4e+26) {
		tmp = x / (y * t);
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = x / (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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-4d+26)) then
        tmp = x / (y * t)
    else if (t_1 <= 0.4d0) then
        tmp = 1.0d0
    else
        tmp = x / (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -4e+26) {
		tmp = x / (y * t);
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = x / (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -4e+26:
		tmp = x / (y * t)
	elif t_1 <= 0.4:
		tmp = 1.0
	else:
		tmp = x / (y * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -4e+26)
		tmp = Float64(x / Float64(y * t));
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -4e+26)
		tmp = x / (y * t);
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = x / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+26], N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.4], 1.0, N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -4.00000000000000019e26

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      9. *-lowering-*.f64 72.2

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \left(0 - t\right) - z, \color{blue}{t \cdot z}\right)} \]

   5. Simplified 72.2%

      \[\leadsto 1 - \frac{x}{\color{blue}{\mathsf{fma}\left(y, \left(0 - t\right) - z, t \cdot z\right)}} \]

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 28.1

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

   8. Simplified 28.1%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 28.1

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

   11. Simplified 28.1%

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

   if -4.00000000000000019e26 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 97.1%

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

   if 0.40000000000000002 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-frac-neg2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 66.4

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

   7. Simplified 66.4%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 62.6

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

   10. Simplified 62.6%

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

   11. Taylor expanded in y around inf

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 37.3

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

   13. Simplified 37.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -4 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{y \cdot t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* y t))))
   (if (<= t_1 -4e+26) t_2 (if (<= t_1 0.4) 1.0 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (y * t);
	double tmp;
	if (t_1 <= -4e+26) {
		tmp = t_2;
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / (y * t)
    if (t_1 <= (-4d+26)) then
        tmp = t_2
    else if (t_1 <= 0.4d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (y * t);
	double tmp;
	if (t_1 <= -4e+26) {
		tmp = t_2;
	} else if (t_1 <= 0.4) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / (y * t)
	tmp = 0
	if t_1 <= -4e+26:
		tmp = t_2
	elif t_1 <= 0.4:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(y * t))
	tmp = 0.0
	if (t_1 <= -4e+26)
		tmp = t_2;
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / (y * t);
	tmp = 0.0;
	if (t_1 <= -4e+26)
		tmp = t_2;
	elseif (t_1 <= 0.4)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+26], t$95$2, If[LessEqual[t$95$1, 0.4], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -4.00000000000000019e26 or 0.40000000000000002 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \color{blue}{\left(0 - t\right)} - z, t \cdot z\right)} \]

      9. *-lowering-*.f64 72.8

         \[\leadsto 1 - \frac{x}{\mathsf{fma}\left(y, \left(0 - t\right) - z, \color{blue}{t \cdot z}\right)} \]

   5. Simplified 72.8%

      \[\leadsto 1 - \frac{x}{\color{blue}{\mathsf{fma}\left(y, \left(0 - t\right) - z, t \cdot z\right)}} \]

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 24.3

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

   8. Simplified 24.3%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 24.3

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

   11. Simplified 24.3%

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

   if -4.00000000000000019e26 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 97.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-189}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.55e-189)
   (+ 1.0 (/ x (* z (- y t))))
   (if (<= t 6.5e-54) (- 1.0 (/ x (* y y))) (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.55e-189) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (t <= 6.5e-54) {
		tmp = 1.0 - (x / (y * y));
	} 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 (t <= (-1.55d-189)) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else if (t <= 6.5d-54) then
        tmp = 1.0d0 - (x / (y * y))
    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 (t <= -1.55e-189) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (t <= 6.5e-54) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.55e-189:
		tmp = 1.0 + (x / (z * (y - t)))
	elif t <= 6.5e-54:
		tmp = 1.0 - (x / (y * y))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.55e-189)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	elseif (t <= 6.5e-54)
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	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 (t <= -1.55e-189)
		tmp = 1.0 + (x / (z * (y - t)));
	elseif (t <= 6.5e-54)
		tmp = 1.0 - (x / (y * y));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.55e-189], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e-54], N[(1.0 - N[(x / N[(y * y), $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 t < -1.55e-189

   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 inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 76.9

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

   5. Simplified 76.9%

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

   if -1.55e-189 < t < 6.49999999999999991e-54

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{{y}^{2}}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 69.1

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

   5. Simplified 69.1%

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

   if 6.49999999999999991e-54 < 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 t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. /-lowering-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-lowering-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. unsub-neg N/A

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

      19. mul-1-neg N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 94.7

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

   5. Simplified 94.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-189}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-55}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-55}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.4e-55)
   (- 1.0 (/ x (* z t)))
   (if (<= t 6.6e-55) (- 1.0 (/ x (* y y))) (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.4e-55) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 6.6e-55) {
		tmp = 1.0 - (x / (y * y));
	} 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 (t <= (-1.4d-55)) then
        tmp = 1.0d0 - (x / (z * t))
    else if (t <= 6.6d-55) then
        tmp = 1.0d0 - (x / (y * y))
    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 (t <= -1.4e-55) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 6.6e-55) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.4e-55:
		tmp = 1.0 - (x / (z * t))
	elif t <= 6.6e-55:
		tmp = 1.0 - (x / (y * y))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.4e-55)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	elseif (t <= 6.6e-55)
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	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 (t <= -1.4e-55)
		tmp = 1.0 - (x / (z * t));
	elseif (t <= 6.6e-55)
		tmp = 1.0 - (x / (y * y));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.4e-55], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.6e-55], N[(1.0 - N[(x / N[(y * y), $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 t < -1.39999999999999992e-55

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 71.5

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

   5. Simplified 71.5%

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

   if -1.39999999999999992e-55 < t < 6.5999999999999999e-55

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{{y}^{2}}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 71.7

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

   5. Simplified 71.7%

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

   if 6.5999999999999999e-55 < 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 t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. /-lowering-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-lowering-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. unsub-neg N/A

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

      19. mul-1-neg N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 94.7

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

   5. Simplified 94.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-55}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-55}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-105}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-79}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.7e-105)
   (- 1.0 (/ x (* y y)))
   (if (<= y 3.2e-79) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.7e-105) {
		tmp = 1.0 - (x / (y * y));
	} else if (y <= 3.2e-79) {
		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 <= (-2.7d-105)) then
        tmp = 1.0d0 - (x / (y * y))
    else if (y <= 3.2d-79) 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 <= -2.7e-105) {
		tmp = 1.0 - (x / (y * y));
	} else if (y <= 3.2e-79) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.7e-105:
		tmp = 1.0 - (x / (y * y))
	elif y <= 3.2e-79:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.7e-105)
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	elseif (y <= 3.2e-79)
		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 <= -2.7e-105)
		tmp = 1.0 - (x / (y * y));
	elseif (y <= 3.2e-79)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.7e-105], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-79], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -2.69999999999999993e-105

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{{y}^{2}}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.9

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

   5. Simplified 84.9%

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

   if -2.69999999999999993e-105 < y < 3.19999999999999988e-79

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 82.7

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

   5. Simplified 82.7%

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

   if 3.19999999999999988e-79 < 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 x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 86.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-105}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-79}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.0% accurate, 26.0× 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 99.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 71.3%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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)))))