Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + x \cdot y\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* x y)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (x * y)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (x * y)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(x * y)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (x * y)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \left(\frac{x}{2} + x \cdot y\right) + z \]
4. Add Preprocessing

Alternative 2: 61.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-63}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-152}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-150}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-45}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e+62)
   (* x y)
   (if (<= y -1.12e-63)
     z
     (if (<= y -8.2e-152)
       (* x 0.5)
       (if (<= y 1.05e-150)
         z
         (if (<= y 1.15e-106)
           (* x 0.5)
           (if (<= y 8e-45) z (if (<= y 0.5) (* x 0.5) (* x y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+62) {
		tmp = x * y;
	} else if (y <= -1.12e-63) {
		tmp = z;
	} else if (y <= -8.2e-152) {
		tmp = x * 0.5;
	} else if (y <= 1.05e-150) {
		tmp = z;
	} else if (y <= 1.15e-106) {
		tmp = x * 0.5;
	} else if (y <= 8e-45) {
		tmp = z;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.2d+62)) then
        tmp = x * y
    else if (y <= (-1.12d-63)) then
        tmp = z
    else if (y <= (-8.2d-152)) then
        tmp = x * 0.5d0
    else if (y <= 1.05d-150) then
        tmp = z
    else if (y <= 1.15d-106) then
        tmp = x * 0.5d0
    else if (y <= 8d-45) then
        tmp = z
    else if (y <= 0.5d0) then
        tmp = x * 0.5d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+62) {
		tmp = x * y;
	} else if (y <= -1.12e-63) {
		tmp = z;
	} else if (y <= -8.2e-152) {
		tmp = x * 0.5;
	} else if (y <= 1.05e-150) {
		tmp = z;
	} else if (y <= 1.15e-106) {
		tmp = x * 0.5;
	} else if (y <= 8e-45) {
		tmp = z;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2e+62:
		tmp = x * y
	elif y <= -1.12e-63:
		tmp = z
	elif y <= -8.2e-152:
		tmp = x * 0.5
	elif y <= 1.05e-150:
		tmp = z
	elif y <= 1.15e-106:
		tmp = x * 0.5
	elif y <= 8e-45:
		tmp = z
	elif y <= 0.5:
		tmp = x * 0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2e+62)
		tmp = Float64(x * y);
	elseif (y <= -1.12e-63)
		tmp = z;
	elseif (y <= -8.2e-152)
		tmp = Float64(x * 0.5);
	elseif (y <= 1.05e-150)
		tmp = z;
	elseif (y <= 1.15e-106)
		tmp = Float64(x * 0.5);
	elseif (y <= 8e-45)
		tmp = z;
	elseif (y <= 0.5)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.2e+62)
		tmp = x * y;
	elseif (y <= -1.12e-63)
		tmp = z;
	elseif (y <= -8.2e-152)
		tmp = x * 0.5;
	elseif (y <= 1.05e-150)
		tmp = z;
	elseif (y <= 1.15e-106)
		tmp = x * 0.5;
	elseif (y <= 8e-45)
		tmp = z;
	elseif (y <= 0.5)
		tmp = x * 0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2e+62], N[(x * y), $MachinePrecision], If[LessEqual[y, -1.12e-63], z, If[LessEqual[y, -8.2e-152], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 1.05e-150], z, If[LessEqual[y, 1.15e-106], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 8e-45], z, If[LessEqual[y, 0.5], N[(x * 0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e62 or 0.5 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 59.0%

         \[\leadsto z - x \cdot \color{blue}{\frac{-0.5 \cdot -0.5 - y \cdot y}{-0.5 + y}} \]

      2. associate-*r/ 50.6%

         \[\leadsto z - \color{blue}{\frac{x \cdot \left(-0.5 \cdot -0.5 - y \cdot y\right)}{-0.5 + y}} \]

      3. metadata-eval 50.6%

         \[\leadsto z - \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{-0.5 + y} \]

      4. pow2 50.6%

         \[\leadsto z - \frac{x \cdot \left(0.25 - \color{blue}{{y}^{2}}\right)}{-0.5 + y} \]

      5. +-commutative 50.6%

         \[\leadsto z - \frac{x \cdot \left(0.25 - {y}^{2}\right)}{\color{blue}{y + -0.5}} \]

   6. Applied egg-rr 50.6%

      \[\leadsto z - \color{blue}{\frac{x \cdot \left(0.25 - {y}^{2}\right)}{y + -0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 50.6%

         \[\leadsto z - \frac{\color{blue}{\left(0.25 - {y}^{2}\right) \cdot x}}{y + -0.5} \]

      2. associate-/l* 57.6%

         \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   8. Simplified 57.6%

      \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   9. Taylor expanded in y around inf 68.8%

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

   if -1.2e62 < y < -1.12000000000000002e-63 or -8.2000000000000002e-152 < y < 1.0500000000000001e-150 or 1.15e-106 < y < 7.99999999999999987e-45

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 100.0%

         \[\leadsto z - x \cdot \color{blue}{\frac{-0.5 \cdot -0.5 - y \cdot y}{-0.5 + y}} \]

      2. associate-*r/ 99.1%

         \[\leadsto z - \color{blue}{\frac{x \cdot \left(-0.5 \cdot -0.5 - y \cdot y\right)}{-0.5 + y}} \]

      3. metadata-eval 99.1%

         \[\leadsto z - \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{-0.5 + y} \]

      4. pow2 99.1%

         \[\leadsto z - \frac{x \cdot \left(0.25 - \color{blue}{{y}^{2}}\right)}{-0.5 + y} \]

      5. +-commutative 99.1%

         \[\leadsto z - \frac{x \cdot \left(0.25 - {y}^{2}\right)}{\color{blue}{y + -0.5}} \]

   6. Applied egg-rr 99.1%

      \[\leadsto z - \color{blue}{\frac{x \cdot \left(0.25 - {y}^{2}\right)}{y + -0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto z - \frac{\color{blue}{\left(0.25 - {y}^{2}\right) \cdot x}}{y + -0.5} \]

      2. associate-/l* 99.9%

         \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   8. Simplified 99.9%

      \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   9. Taylor expanded in z around inf 62.3%

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

   if -1.12000000000000002e-63 < y < -8.2000000000000002e-152 or 1.0500000000000001e-150 < y < 1.15e-106 or 7.99999999999999987e-45 < y < 0.5

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 70.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 70.0%

         \[\leadsto x \cdot \color{blue}{\left(y + 0.5\right)} \]

   7. Simplified 70.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right)} \]

   8. Taylor expanded in y around 0 68.9%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 68.9%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   10. Simplified 68.9%

       \[\leadsto \color{blue}{x \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-63}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-152}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-150}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-45}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-16} \lor \neg \left(z \leq 1.05 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e-16) (not (<= z 1.05e-23))) (+ (* x y) z) (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e-16) || !(z <= 1.05e-23)) {
		tmp = (x * y) + z;
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.8d-16)) .or. (.not. (z <= 1.05d-23))) then
        tmp = (x * y) + z
    else
        tmp = x * (y + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e-16) || !(z <= 1.05e-23)) {
		tmp = (x * y) + z;
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.8e-16) or not (z <= 1.05e-23):
		tmp = (x * y) + z
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e-16) || !(z <= 1.05e-23))
		tmp = Float64(Float64(x * y) + z);
	else
		tmp = Float64(x * Float64(y + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.8e-16) || ~((z <= 1.05e-23)))
		tmp = (x * y) + z;
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e-16], N[Not[LessEqual[z, 1.05e-23]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000012e-16 or 1.05e-23 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 90.5%

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

      1. associate-*r* 90.5%

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

      2. neg-mul-1 90.5%

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

   7. Simplified 90.5%

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

      1. cancel-sign-sub 90.5%

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

      2. +-commutative 90.5%

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

      3. *-commutative 90.5%

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

   9. Applied egg-rr 90.5%

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

   if -3.80000000000000012e-16 < z < 1.05e-23

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 87.3%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 87.3%

         \[\leadsto x \cdot \color{blue}{\left(y + 0.5\right)} \]

   7. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-16} \lor \neg \left(z \leq 1.05 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0045\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.5) (not (<= y 0.0045))) (+ (* x y) z) (- z (* x -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.0045)) {
		tmp = (x * y) + z;
	} else {
		tmp = z - (x * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-0.5d0)) .or. (.not. (y <= 0.0045d0))) then
        tmp = (x * y) + z
    else
        tmp = z - (x * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.0045)) {
		tmp = (x * y) + z;
	} else {
		tmp = z - (x * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.5) or not (y <= 0.0045):
		tmp = (x * y) + z
	else:
		tmp = z - (x * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.5) || !(y <= 0.0045))
		tmp = Float64(Float64(x * y) + z);
	else
		tmp = Float64(z - Float64(x * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.5) || ~((y <= 0.0045)))
		tmp = (x * y) + z;
	else
		tmp = z - (x * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.5], N[Not[LessEqual[y, 0.0045]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision], N[(z - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 0.00449999999999999966 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. neg-mul-1 98.0%

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

   7. Simplified 98.0%

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

      1. cancel-sign-sub 98.0%

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

      2. +-commutative 98.0%

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

      3. *-commutative 98.0%

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

   9. Applied egg-rr 98.0%

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

   if -0.5 < y < 0.00449999999999999966

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 99.9%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.0%

      \[\leadsto z - \color{blue}{-0.5 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto z - \color{blue}{x \cdot -0.5} \]

   7. Simplified 98.0%

      \[\leadsto z - \color{blue}{x \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0045\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+114}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.06e+114) z (if (<= z 6e+169) (* x (+ y 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.06e+114) {
		tmp = z;
	} else if (z <= 6e+169) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.06d+114)) then
        tmp = z
    else if (z <= 6d+169) then
        tmp = x * (y + 0.5d0)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.06e+114) {
		tmp = z;
	} else if (z <= 6e+169) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.06e+114:
		tmp = z
	elif z <= 6e+169:
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.06e+114)
		tmp = z;
	elseif (z <= 6e+169)
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.06e+114)
		tmp = z;
	elseif (z <= 6e+169)
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.06e+114], z, If[LessEqual[z, 6e+169], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05999999999999993e114 or 5.9999999999999999e169 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 81.0%

         \[\leadsto z - x \cdot \color{blue}{\frac{-0.5 \cdot -0.5 - y \cdot y}{-0.5 + y}} \]

      2. associate-*r/ 77.5%

         \[\leadsto z - \color{blue}{\frac{x \cdot \left(-0.5 \cdot -0.5 - y \cdot y\right)}{-0.5 + y}} \]

      3. metadata-eval 77.5%

         \[\leadsto z - \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{-0.5 + y} \]

      4. pow2 77.5%

         \[\leadsto z - \frac{x \cdot \left(0.25 - \color{blue}{{y}^{2}}\right)}{-0.5 + y} \]

      5. +-commutative 77.5%

         \[\leadsto z - \frac{x \cdot \left(0.25 - {y}^{2}\right)}{\color{blue}{y + -0.5}} \]

   6. Applied egg-rr 77.5%

      \[\leadsto z - \color{blue}{\frac{x \cdot \left(0.25 - {y}^{2}\right)}{y + -0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 77.5%

         \[\leadsto z - \frac{\color{blue}{\left(0.25 - {y}^{2}\right) \cdot x}}{y + -0.5} \]

      2. associate-/l* 80.2%

         \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   8. Simplified 80.2%

      \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   9. Taylor expanded in z around inf 86.4%

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

   if -1.05999999999999993e114 < z < 5.9999999999999999e169

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.2%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 78.2%

         \[\leadsto x \cdot \color{blue}{\left(y + 0.5\right)} \]

   7. Simplified 78.2%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+114}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-26}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5e-11) z (if (<= z 2.35e-26) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-11) {
		tmp = z;
	} else if (z <= 2.35e-26) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.5d-11)) then
        tmp = z
    else if (z <= 2.35d-26) then
        tmp = x * 0.5d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-11) {
		tmp = z;
	} else if (z <= 2.35e-26) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.5e-11:
		tmp = z
	elif z <= 2.35e-26:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5e-11)
		tmp = z;
	elseif (z <= 2.35e-26)
		tmp = Float64(x * 0.5);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.5e-11)
		tmp = z;
	elseif (z <= 2.35e-26)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5e-11], z, If[LessEqual[z, 2.35e-26], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -5.49999999999999975e-11 or 2.34999999999999995e-26 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

      4. distribute-frac-neg 100.0%

         \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

      5. distribute-rgt-neg-out 100.0%

         \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

      6. unsub-neg 100.0%

         \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

      9. neg-mul-1 100.0%

         \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

      10. associate-/l* 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

      11. associate-/r/ 100.0%

          \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

      12. distribute-rgt-out-- 100.0%

          \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 81.6%

         \[\leadsto z - x \cdot \color{blue}{\frac{-0.5 \cdot -0.5 - y \cdot y}{-0.5 + y}} \]

      2. associate-*r/ 76.3%

         \[\leadsto z - \color{blue}{\frac{x \cdot \left(-0.5 \cdot -0.5 - y \cdot y\right)}{-0.5 + y}} \]

      3. metadata-eval 76.3%

         \[\leadsto z - \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{-0.5 + y} \]

      4. pow2 76.3%

         \[\leadsto z - \frac{x \cdot \left(0.25 - \color{blue}{{y}^{2}}\right)}{-0.5 + y} \]

      5. +-commutative 76.3%

         \[\leadsto z - \frac{x \cdot \left(0.25 - {y}^{2}\right)}{\color{blue}{y + -0.5}} \]

   6. Applied egg-rr 76.3%

      \[\leadsto z - \color{blue}{\frac{x \cdot \left(0.25 - {y}^{2}\right)}{y + -0.5}} \]
   7. Step-by-step derivation

      1. *-commutative 76.3%

         \[\leadsto z - \frac{\color{blue}{\left(0.25 - {y}^{2}\right) \cdot x}}{y + -0.5} \]

      2. associate-/l* 80.9%

         \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   8. Simplified 80.9%

      \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

   9. Taylor expanded in z around inf 66.8%

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

   if -5.49999999999999975e-11 < z < 2.34999999999999995e-26

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 87.3%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 87.3%

         \[\leadsto x \cdot \color{blue}{\left(y + 0.5\right)} \]

   7. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right)} \]

   8. Taylor expanded in y around 0 45.9%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 45.9%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   10. Simplified 45.9%

       \[\leadsto \color{blue}{x \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-26}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ z + x \cdot \left(y - -0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ z (* x (- y -0.5))))

C

double code(double x, double y, double z) {
	return z + (x * (y - -0.5));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z + (x * (y - (-0.5d0)))
end function

Java

public static double code(double x, double y, double z) {
	return z + (x * (y - -0.5));
}

Python

def code(x, y, z):
	return z + (x * (y - -0.5))

Julia

function code(x, y, z)
	return Float64(z + Float64(x * Float64(y - -0.5)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z + (x * (y - -0.5));
end

Wolfram

code[x_, y_, z_] := N[(z + N[(x * N[(y - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

   2. remove-double-neg 100.0%

      \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

   3. distribute-neg-in 100.0%

      \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

   4. distribute-frac-neg 100.0%

      \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

   5. distribute-rgt-neg-out 100.0%

      \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

   6. unsub-neg 100.0%

      \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

   7. distribute-rgt-neg-out 100.0%

      \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

   8. unsub-neg 100.0%

      \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

   9. neg-mul-1 100.0%

      \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

   10. associate-/l* 99.9%

       \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

   11. associate-/r/ 100.0%

       \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

   12. distribute-rgt-out-- 100.0%

       \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

   13. metadata-eval 100.0%

       \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y - -0.5\right) \]
6. Add Preprocessing

Alternative 8: 40.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{z + \left(\frac{x}{2} + y \cdot x\right)} \]

   2. remove-double-neg 100.0%

      \[\leadsto z + \color{blue}{\left(-\left(-\left(\frac{x}{2} + y \cdot x\right)\right)\right)} \]

   3. distribute-neg-in 100.0%

      \[\leadsto z + \left(-\color{blue}{\left(\left(-\frac{x}{2}\right) + \left(-y \cdot x\right)\right)}\right) \]

   4. distribute-frac-neg 100.0%

      \[\leadsto z + \left(-\left(\color{blue}{\frac{-x}{2}} + \left(-y \cdot x\right)\right)\right) \]

   5. distribute-rgt-neg-out 100.0%

      \[\leadsto z + \left(-\left(\frac{-x}{2} + \color{blue}{y \cdot \left(-x\right)}\right)\right) \]

   6. unsub-neg 100.0%

      \[\leadsto \color{blue}{z - \left(\frac{-x}{2} + y \cdot \left(-x\right)\right)} \]

   7. distribute-rgt-neg-out 100.0%

      \[\leadsto z - \left(\frac{-x}{2} + \color{blue}{\left(-y \cdot x\right)}\right) \]

   8. unsub-neg 100.0%

      \[\leadsto z - \color{blue}{\left(\frac{-x}{2} - y \cdot x\right)} \]

   9. neg-mul-1 100.0%

      \[\leadsto z - \left(\frac{\color{blue}{-1 \cdot x}}{2} - y \cdot x\right) \]

   10. associate-/l* 99.9%

       \[\leadsto z - \left(\color{blue}{\frac{-1}{\frac{2}{x}}} - y \cdot x\right) \]

   11. associate-/r/ 100.0%

       \[\leadsto z - \left(\color{blue}{\frac{-1}{2} \cdot x} - y \cdot x\right) \]

   12. distribute-rgt-out-- 100.0%

       \[\leadsto z - \color{blue}{x \cdot \left(\frac{-1}{2} - y\right)} \]

   13. metadata-eval 100.0%

       \[\leadsto z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{z - x \cdot \left(-0.5 - y\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-- 82.2%

      \[\leadsto z - x \cdot \color{blue}{\frac{-0.5 \cdot -0.5 - y \cdot y}{-0.5 + y}} \]

   2. associate-*r/ 78.2%

      \[\leadsto z - \color{blue}{\frac{x \cdot \left(-0.5 \cdot -0.5 - y \cdot y\right)}{-0.5 + y}} \]

   3. metadata-eval 78.2%

      \[\leadsto z - \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{-0.5 + y} \]

   4. pow2 78.2%

      \[\leadsto z - \frac{x \cdot \left(0.25 - \color{blue}{{y}^{2}}\right)}{-0.5 + y} \]

   5. +-commutative 78.2%

      \[\leadsto z - \frac{x \cdot \left(0.25 - {y}^{2}\right)}{\color{blue}{y + -0.5}} \]

6. Applied egg-rr 78.2%

   \[\leadsto z - \color{blue}{\frac{x \cdot \left(0.25 - {y}^{2}\right)}{y + -0.5}} \]
7. Step-by-step derivation

   1. *-commutative 78.2%

      \[\leadsto z - \frac{\color{blue}{\left(0.25 - {y}^{2}\right) \cdot x}}{y + -0.5} \]

   2. associate-/l* 81.5%

      \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

8. Simplified 81.5%

   \[\leadsto z - \color{blue}{\frac{0.25 - {y}^{2}}{\frac{y + -0.5}{x}}} \]

9. Taylor expanded in z around inf 43.2%

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

10. Final simplification 43.2%

    \[\leadsto z \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))