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

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 7

Speedup: N/A×

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, 1.3× 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. *-commutative 100.0%

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

   8. cancel-sign-sub-inv 100.0%

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

   9. neg-mul-1 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-/l* 100.0%

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

   12. distribute-lft-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 2: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot y + \frac{x}{2}\\ \mathbf{if}\;t\_0 \leq -1.5 \cdot 10^{-62} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+90}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* x y) (/ x 2.0))))
   (if (or (<= t_0 -1.5e-62) (not (<= t_0 5e+90))) (* x (+ y 0.5)) z)))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) + (x / 2.0);
	double tmp;
	if ((t_0 <= -1.5e-62) || !(t_0 <= 5e+90)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) + (x / 2.0d0)
    if ((t_0 <= (-1.5d-62)) .or. (.not. (t_0 <= 5d+90))) 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 t_0 = (x * y) + (x / 2.0);
	double tmp;
	if ((t_0 <= -1.5e-62) || !(t_0 <= 5e+90)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * y) + (x / 2.0)
	tmp = 0
	if (t_0 <= -1.5e-62) or not (t_0 <= 5e+90):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) + Float64(x / 2.0))
	tmp = 0.0
	if ((t_0 <= -1.5e-62) || !(t_0 <= 5e+90))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * y) + (x / 2.0);
	tmp = 0.0;
	if ((t_0 <= -1.5e-62) || ~((t_0 <= 5e+90)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] + N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1.5e-62], N[Not[LessEqual[t$95$0, 5e+90]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -1.5000000000000001e-62 or 5.0000000000000004e90 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x))

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around 0 81.2%

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

   if -1.5000000000000001e-62 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 5.0000000000000004e90

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around inf 72.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{x}{2} \leq -1.5 \cdot 10^{-62} \lor \neg \left(x \cdot y + \frac{x}{2} \leq 5 \cdot 10^{+90}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-123}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-143}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+52}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6e+35)
   (* x y)
   (if (<= y -1.4e-123)
     z
     (if (<= y 2.75e-143) (* x 0.5) (if (<= y 4.3e+52) z (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+35) {
		tmp = x * y;
	} else if (y <= -1.4e-123) {
		tmp = z;
	} else if (y <= 2.75e-143) {
		tmp = x * 0.5;
	} else if (y <= 4.3e+52) {
		tmp = z;
	} 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 <= (-4.6d+35)) then
        tmp = x * y
    else if (y <= (-1.4d-123)) then
        tmp = z
    else if (y <= 2.75d-143) then
        tmp = x * 0.5d0
    else if (y <= 4.3d+52) then
        tmp = z
    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 <= -4.6e+35) {
		tmp = x * y;
	} else if (y <= -1.4e-123) {
		tmp = z;
	} else if (y <= 2.75e-143) {
		tmp = x * 0.5;
	} else if (y <= 4.3e+52) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.6e+35:
		tmp = x * y
	elif y <= -1.4e-123:
		tmp = z
	elif y <= 2.75e-143:
		tmp = x * 0.5
	elif y <= 4.3e+52:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.6e+35)
		tmp = Float64(x * y);
	elseif (y <= -1.4e-123)
		tmp = z;
	elseif (y <= 2.75e-143)
		tmp = Float64(x * 0.5);
	elseif (y <= 4.3e+52)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.6e+35)
		tmp = x * y;
	elseif (y <= -1.4e-123)
		tmp = z;
	elseif (y <= 2.75e-143)
		tmp = x * 0.5;
	elseif (y <= 4.3e+52)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.6e+35], N[(x * y), $MachinePrecision], If[LessEqual[y, -1.4e-123], z, If[LessEqual[y, 2.75e-143], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 4.3e+52], z, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5999999999999996e35 or 4.3e52 < 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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 75.6%

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

   if -4.5999999999999996e35 < y < -1.3999999999999999e-123 or 2.75000000000000021e-143 < y < 4.3e52

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around inf 65.1%

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

   if -1.3999999999999999e-123 < y < 2.75000000000000021e-143

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around 0 61.8%

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

   6. Taylor expanded in y around 0 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

Alternative 4: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 1.75 \cdot 10^{-9}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.5) (not (<= y 1.75e-9))) (+ z (* x y)) (- z (* x -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 1.75e-9)) {
		tmp = z + (x * y);
	} 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 <= 1.75d-9))) then
        tmp = z + (x * y)
    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 <= 1.75e-9)) {
		tmp = z + (x * y);
	} else {
		tmp = z - (x * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.5) or not (y <= 1.75e-9):
		tmp = z + (x * y)
	else:
		tmp = z - (x * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.5) || !(y <= 1.75e-9))
		tmp = Float64(z + Float64(x * y));
	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 <= 1.75e-9)))
		tmp = z + (x * y);
	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, 1.75e-9]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 1.75e-9 < 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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-rgt-neg-out 99.1%

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

   7. Simplified 99.1%

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

      1. sub-neg 99.1%

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

      2. distribute-rgt-neg-out 99.1%

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

      3. remove-double-neg 99.1%

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

      4. +-commutative 99.1%

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

   9. Applied egg-rr 99.1%

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

   if -0.5 < y < 1.75e-9

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 1.75 \cdot 10^{-9}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+25} \lor \neg \left(z \leq 1.55 \cdot 10^{+18}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.5e+25) (not (<= z 1.55e+18))) (+ z (* x y)) (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e+25) || !(z <= 1.55e+18)) {
		tmp = z + (x * y);
	} 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 <= (-2.5d+25)) .or. (.not. (z <= 1.55d+18))) then
        tmp = z + (x * y)
    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 <= -2.5e+25) || !(z <= 1.55e+18)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.5e+25) or not (z <= 1.55e+18):
		tmp = z + (x * y)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.5e+25) || !(z <= 1.55e+18))
		tmp = Float64(z + Float64(x * y));
	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 <= -2.5e+25) || ~((z <= 1.55e+18)))
		tmp = z + (x * y);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.5e+25], N[Not[LessEqual[z, 1.55e+18]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000012e25 or 1.55e18 < 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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 94.3%

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

      1. mul-1-neg 94.3%

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

      2. distribute-rgt-neg-out 94.3%

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

   7. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. distribute-rgt-neg-out 94.3%

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

      3. remove-double-neg 94.3%

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

      4. +-commutative 94.3%

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

   9. Applied egg-rr 94.3%

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

   if -2.50000000000000012e25 < z < 1.55e18

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around 0 83.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+25} \lor \neg \left(z \leq 1.55 \cdot 10^{+18}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-84}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.7e+25) z (if (<= z 5.5e-84) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.7e+25) {
		tmp = z;
	} else if (z <= 5.5e-84) {
		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 <= (-3.7d+25)) then
        tmp = z
    else if (z <= 5.5d-84) 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 <= -3.7e+25) {
		tmp = z;
	} else if (z <= 5.5e-84) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.7e+25:
		tmp = z
	elif z <= 5.5e-84:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.7e+25)
		tmp = z;
	elseif (z <= 5.5e-84)
		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 <= -3.7e+25)
		tmp = z;
	elseif (z <= 5.5e-84)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.7e+25], z, If[LessEqual[z, 5.5e-84], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6999999999999999e25 or 5.50000000000000019e-84 < 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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around inf 60.7%

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

   if -3.6999999999999999e25 < z < 5.50000000000000019e-84

   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. *-commutative 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. neg-mul-1 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. distribute-lft-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 z around 0 86.0%

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

   6. Taylor expanded in y around 0 47.8%

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

      1. *-commutative 47.8%

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

   8. Simplified 47.8%

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

Alternative 7: 40.5% 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. *-commutative 100.0%

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

   8. cancel-sign-sub-inv 100.0%

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

   9. neg-mul-1 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-/l* 100.0%

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

   12. distribute-lft-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 z around inf 40.2%

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

Reproduce

herbie shell --seed 2024193 
(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))