Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3

Percentage Accurate: 98.0% → 100.0%

Time: 5.9s

Alternatives: 7

Speedup: 1.3×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y - z, z\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(x, Float64(y - z), z)
end

Wolfram

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

Derivation

1. Initial program 98.4%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Step-by-step derivation

   1. *-commutative 98.4%

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

   2. distribute-lft-out-- 98.4%

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

   3. *-rgt-identity 98.4%

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

   4. cancel-sign-sub-inv 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-+r+ 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. distribute-rgt-out 100.0%

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

   10. fma-define 100.0%

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

   11. +-commutative 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, y - z, z\right) \]
6. Add Preprocessing

Alternative 2: 54.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1.15:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-214}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-295}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{-302}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-192}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-37}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= y -1.15)
     (* x y)
     (if (<= y -6e-214)
       z
       (if (<= y -4e-273)
         t_0
         (if (<= y -3.1e-295)
           z
           (if (<= y 3.15e-302)
             t_0
             (if (<= y 6.2e-192)
               z
               (if (<= y 1.26e-134) t_0 (if (<= y 5e-37) z (* x y)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (y <= -1.15) {
		tmp = x * y;
	} else if (y <= -6e-214) {
		tmp = z;
	} else if (y <= -4e-273) {
		tmp = t_0;
	} else if (y <= -3.1e-295) {
		tmp = z;
	} else if (y <= 3.15e-302) {
		tmp = t_0;
	} else if (y <= 6.2e-192) {
		tmp = z;
	} else if (y <= 1.26e-134) {
		tmp = t_0;
	} else if (y <= 5e-37) {
		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) :: t_0
    real(8) :: tmp
    t_0 = z * -x
    if (y <= (-1.15d0)) then
        tmp = x * y
    else if (y <= (-6d-214)) then
        tmp = z
    else if (y <= (-4d-273)) then
        tmp = t_0
    else if (y <= (-3.1d-295)) then
        tmp = z
    else if (y <= 3.15d-302) then
        tmp = t_0
    else if (y <= 6.2d-192) then
        tmp = z
    else if (y <= 1.26d-134) then
        tmp = t_0
    else if (y <= 5d-37) 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 t_0 = z * -x;
	double tmp;
	if (y <= -1.15) {
		tmp = x * y;
	} else if (y <= -6e-214) {
		tmp = z;
	} else if (y <= -4e-273) {
		tmp = t_0;
	} else if (y <= -3.1e-295) {
		tmp = z;
	} else if (y <= 3.15e-302) {
		tmp = t_0;
	} else if (y <= 6.2e-192) {
		tmp = z;
	} else if (y <= 1.26e-134) {
		tmp = t_0;
	} else if (y <= 5e-37) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if y <= -1.15:
		tmp = x * y
	elif y <= -6e-214:
		tmp = z
	elif y <= -4e-273:
		tmp = t_0
	elif y <= -3.1e-295:
		tmp = z
	elif y <= 3.15e-302:
		tmp = t_0
	elif y <= 6.2e-192:
		tmp = z
	elif y <= 1.26e-134:
		tmp = t_0
	elif y <= 5e-37:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (y <= -1.15)
		tmp = Float64(x * y);
	elseif (y <= -6e-214)
		tmp = z;
	elseif (y <= -4e-273)
		tmp = t_0;
	elseif (y <= -3.1e-295)
		tmp = z;
	elseif (y <= 3.15e-302)
		tmp = t_0;
	elseif (y <= 6.2e-192)
		tmp = z;
	elseif (y <= 1.26e-134)
		tmp = t_0;
	elseif (y <= 5e-37)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (y <= -1.15)
		tmp = x * y;
	elseif (y <= -6e-214)
		tmp = z;
	elseif (y <= -4e-273)
		tmp = t_0;
	elseif (y <= -3.1e-295)
		tmp = z;
	elseif (y <= 3.15e-302)
		tmp = t_0;
	elseif (y <= 6.2e-192)
		tmp = z;
	elseif (y <= 1.26e-134)
		tmp = t_0;
	elseif (y <= 5e-37)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[y, -1.15], N[(x * y), $MachinePrecision], If[LessEqual[y, -6e-214], z, If[LessEqual[y, -4e-273], t$95$0, If[LessEqual[y, -3.1e-295], z, If[LessEqual[y, 3.15e-302], t$95$0, If[LessEqual[y, 6.2e-192], z, If[LessEqual[y, 1.26e-134], t$95$0, If[LessEqual[y, 5e-37], z, N[(x * y), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1499999999999999 or 4.9999999999999997e-37 < y

   1. Initial program 96.9%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.2%

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

   if -1.1499999999999999 < y < -5.99999999999999989e-214 or -4e-273 < y < -3.1000000000000002e-295 or 3.1499999999999998e-302 < y < 6.2000000000000001e-192 or 1.25999999999999989e-134 < y < 4.9999999999999997e-37

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.6%

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

   if -5.99999999999999989e-214 < y < -4e-273 or -3.1000000000000002e-295 < y < 3.1499999999999998e-302 or 6.2000000000000001e-192 < y < 1.25999999999999989e-134

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.3%

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

      1. mul-1-neg 81.3%

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

      2. sub-neg 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in y around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. distribute-lft-neg-out 69.6%

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

      3. *-commutative 69.6%

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

   8. Simplified 69.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-214}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-273}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-295}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{-302}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-192}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-134}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-37}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-52} \lor \neg \left(x \leq 1.05 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.5e-52) (not (<= x 1.05e-14))) (* x (- y z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e-52) || !(x <= 1.05e-14)) {
		tmp = x * (y - z);
	} 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 ((x <= (-5.5d-52)) .or. (.not. (x <= 1.05d-14))) then
        tmp = x * (y - z)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e-52) || !(x <= 1.05e-14)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.5e-52) or not (x <= 1.05e-14):
		tmp = x * (y - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.5e-52) || !(x <= 1.05e-14))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.5e-52) || ~((x <= 1.05e-14)))
		tmp = x * (y - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.5e-52], N[Not[LessEqual[x, 1.05e-14]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e-52 or 1.0499999999999999e-14 < x

   1. Initial program 97.2%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.2%

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

      1. mul-1-neg 96.2%

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

      2. sub-neg 96.2%

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

   5. Simplified 96.2%

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

   if -5.5e-52 < x < 1.0499999999999999e-14

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-52} \lor \neg \left(x \leq 1.05 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.53\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.53))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.53)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (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 ((x <= (-1.0d0)) .or. (.not. (x <= 0.53d0))) then
        tmp = x * (y - z)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.53)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.53):
		tmp = x * (y - z)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.53))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.53)))
		tmp = x * (y - z);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.53]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.53000000000000003 < x

   1. Initial program 96.9%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. sub-neg 99.2%

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

   5. Simplified 99.2%

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

   if -1 < x < 0.53000000000000003

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

      5. neg-sub0 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. *-lft-identity 100.0%

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

      11. associate-+l- 100.0%

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

      12. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 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. *-commutative 99.1%

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

      2. cancel-sign-sub 99.1%

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

      3. +-commutative 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.53\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.44 \lor \neg \left(y \leq 1.3 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.44) (not (<= y 1.3e-28))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.44) || !(y <= 1.3e-28)) {
		tmp = x * y;
	} 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 ((y <= (-0.44d0)) .or. (.not. (y <= 1.3d-28))) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.44) || !(y <= 1.3e-28)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.44) or not (y <= 1.3e-28):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.44) || !(y <= 1.3e-28))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.44) || ~((y <= 1.3e-28)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.44], N[Not[LessEqual[y, 1.3e-28]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -0.440000000000000002 or 1.3e-28 < y

   1. Initial program 96.9%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.2%

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

   if -0.440000000000000002 < y < 1.3e-28

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 52.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.44 \lor \neg \left(y \leq 1.3 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return z + (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 = z + (x * (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Step-by-step derivation

   1. +-commutative 98.4%

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

   2. remove-double-neg 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. neg-sub0 98.4%

      \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

   5. neg-sub0 98.4%

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

   6. *-commutative 98.4%

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

   7. distribute-lft-neg-in 98.4%

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

   8. remove-double-neg 98.4%

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

   9. distribute-rgt-out-- 98.4%

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

   10. *-lft-identity 98.4%

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

   11. associate-+l- 98.4%

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

   12. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 36.0% 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 98.4%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0 34.2%

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

4. Final simplification 34.2%

   \[\leadsto z \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))