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

Percentage Accurate: 98.2% → 100.0%

Time: 3.5s

Alternatives: 7

Speedup: 1.3×

• 21.1% 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: 98.2% 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, 1.3× speedup

Math

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

FPCore

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

C

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

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 + ((y - z) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft1-in 100.0%

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

   4. associate-+r+ 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. neg-mul-1 100.0%

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

   8. associate-*r* 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. 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. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto z + \left(y - z\right) \cdot x \]

Alternative 2: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+181}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-63}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+115} \lor \neg \left(x \leq 5 \cdot 10^{+142}\right) \land x \leq 2.6 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -6.5e+181)
     (* y x)
     (if (<= x -5.6e+15)
       t_0
       (if (<= x -5.2e-23)
         (* y x)
         (if (<= x 4.5e-63)
           z
           (if (<= x 1.9e-26)
             (* y x)
             (if (<= x 1.0)
               z
               (if (or (<= x 3.1e+115)
                       (and (not (<= x 5e+142)) (<= x 2.6e+220)))
                 t_0
                 (* y x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -6.5e+181) {
		tmp = y * x;
	} else if (x <= -5.6e+15) {
		tmp = t_0;
	} else if (x <= -5.2e-23) {
		tmp = y * x;
	} else if (x <= 4.5e-63) {
		tmp = z;
	} else if (x <= 1.9e-26) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = z;
	} else if ((x <= 3.1e+115) || (!(x <= 5e+142) && (x <= 2.6e+220))) {
		tmp = t_0;
	} else {
		tmp = y * x;
	}
	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 (x <= (-6.5d+181)) then
        tmp = y * x
    else if (x <= (-5.6d+15)) then
        tmp = t_0
    else if (x <= (-5.2d-23)) then
        tmp = y * x
    else if (x <= 4.5d-63) then
        tmp = z
    else if (x <= 1.9d-26) then
        tmp = y * x
    else if (x <= 1.0d0) then
        tmp = z
    else if ((x <= 3.1d+115) .or. (.not. (x <= 5d+142)) .and. (x <= 2.6d+220)) then
        tmp = t_0
    else
        tmp = y * x
    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 (x <= -6.5e+181) {
		tmp = y * x;
	} else if (x <= -5.6e+15) {
		tmp = t_0;
	} else if (x <= -5.2e-23) {
		tmp = y * x;
	} else if (x <= 4.5e-63) {
		tmp = z;
	} else if (x <= 1.9e-26) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = z;
	} else if ((x <= 3.1e+115) || (!(x <= 5e+142) && (x <= 2.6e+220))) {
		tmp = t_0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -6.5e+181:
		tmp = y * x
	elif x <= -5.6e+15:
		tmp = t_0
	elif x <= -5.2e-23:
		tmp = y * x
	elif x <= 4.5e-63:
		tmp = z
	elif x <= 1.9e-26:
		tmp = y * x
	elif x <= 1.0:
		tmp = z
	elif (x <= 3.1e+115) or (not (x <= 5e+142) and (x <= 2.6e+220)):
		tmp = t_0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -6.5e+181)
		tmp = Float64(y * x);
	elseif (x <= -5.6e+15)
		tmp = t_0;
	elseif (x <= -5.2e-23)
		tmp = Float64(y * x);
	elseif (x <= 4.5e-63)
		tmp = z;
	elseif (x <= 1.9e-26)
		tmp = Float64(y * x);
	elseif (x <= 1.0)
		tmp = z;
	elseif ((x <= 3.1e+115) || (!(x <= 5e+142) && (x <= 2.6e+220)))
		tmp = t_0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (x <= -6.5e+181)
		tmp = y * x;
	elseif (x <= -5.6e+15)
		tmp = t_0;
	elseif (x <= -5.2e-23)
		tmp = y * x;
	elseif (x <= 4.5e-63)
		tmp = z;
	elseif (x <= 1.9e-26)
		tmp = y * x;
	elseif (x <= 1.0)
		tmp = z;
	elseif ((x <= 3.1e+115) || (~((x <= 5e+142)) && (x <= 2.6e+220)))
		tmp = t_0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[x, -6.5e+181], N[(y * x), $MachinePrecision], If[LessEqual[x, -5.6e+15], t$95$0, If[LessEqual[x, -5.2e-23], N[(y * x), $MachinePrecision], If[LessEqual[x, 4.5e-63], z, If[LessEqual[x, 1.9e-26], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.0], z, If[Or[LessEqual[x, 3.1e+115], And[N[Not[LessEqual[x, 5e+142]], $MachinePrecision], LessEqual[x, 2.6e+220]]], t$95$0, N[(y * x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e181 or -5.6e15 < x < -5.2e-23 or 4.5e-63 < x < 1.90000000000000007e-26 or 3.10000000000000005e115 < x < 5.0000000000000001e142 or 2.59999999999999994e220 < x

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 78.8%

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

   if -6.5e181 < x < -5.6e15 or 1 < x < 3.10000000000000005e115 or 5.0000000000000001e142 < x < 2.59999999999999994e220

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. 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. Taylor expanded in x around inf 96.5%

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

   5. Taylor expanded in y around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. distribute-rgt-neg-out 66.5%

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

   7. Simplified 66.5%

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

   if -5.2e-23 < x < 4.5e-63 or 1.90000000000000007e-26 < x < 1

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in x around 0 80.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+181}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-63}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+115} \lor \neg \left(x \leq 5 \cdot 10^{+142}\right) \land x \leq 2.6 \cdot 10^{+220}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-64}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 10000000:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) x)))
   (if (<= x -1.9e-24)
     t_0
     (if (<= x 5.6e-64)
       z
       (if (<= x 1.9e-26)
         (* y x)
         (if (<= x 10000000.0) (* z (- 1.0 x)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -1.9e-24) {
		tmp = t_0;
	} else if (x <= 5.6e-64) {
		tmp = z;
	} else if (x <= 1.9e-26) {
		tmp = y * x;
	} else if (x <= 10000000.0) {
		tmp = z * (1.0 - x);
	} else {
		tmp = t_0;
	}
	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 = (y - z) * x
    if (x <= (-1.9d-24)) then
        tmp = t_0
    else if (x <= 5.6d-64) then
        tmp = z
    else if (x <= 1.9d-26) then
        tmp = y * x
    else if (x <= 10000000.0d0) then
        tmp = z * (1.0d0 - x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -1.9e-24) {
		tmp = t_0;
	} else if (x <= 5.6e-64) {
		tmp = z;
	} else if (x <= 1.9e-26) {
		tmp = y * x;
	} else if (x <= 10000000.0) {
		tmp = z * (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) * x
	tmp = 0
	if x <= -1.9e-24:
		tmp = t_0
	elif x <= 5.6e-64:
		tmp = z
	elif x <= 1.9e-26:
		tmp = y * x
	elif x <= 10000000.0:
		tmp = z * (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * x)
	tmp = 0.0
	if (x <= -1.9e-24)
		tmp = t_0;
	elseif (x <= 5.6e-64)
		tmp = z;
	elseif (x <= 1.9e-26)
		tmp = Float64(y * x);
	elseif (x <= 10000000.0)
		tmp = Float64(z * Float64(1.0 - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) * x;
	tmp = 0.0;
	if (x <= -1.9e-24)
		tmp = t_0;
	elseif (x <= 5.6e-64)
		tmp = z;
	elseif (x <= 1.9e-26)
		tmp = y * x;
	elseif (x <= 10000000.0)
		tmp = z * (1.0 - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.9e-24], t$95$0, If[LessEqual[x, 5.6e-64], z, If[LessEqual[x, 1.9e-26], N[(y * x), $MachinePrecision], If[LessEqual[x, 10000000.0], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.90000000000000013e-24 or 1e7 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. 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. Taylor expanded in x around inf 99.8%

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

   if -1.90000000000000013e-24 < x < 5.60000000000000008e-64

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in x around 0 80.9%

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

   if 5.60000000000000008e-64 < x < 1.90000000000000007e-26

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 88.1%

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

   if 1.90000000000000007e-26 < x < 1e7

   1. Initial program 99.8%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{z \cdot \left(1 - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-24}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-64}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 10000000:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -24000000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -24000000000.0) (not (<= x 1.0))) (* (- y z) x) (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -24000000000.0) || !(x <= 1.0)) {
		tmp = (y - z) * x;
	} else {
		tmp = z + (y * x);
	}
	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 <= (-24000000000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (y - z) * x
    else
        tmp = z + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -24000000000.0) || !(x <= 1.0)) {
		tmp = (y - z) * x;
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -24000000000.0) or not (x <= 1.0):
		tmp = (y - z) * x
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -24000000000.0) || !(x <= 1.0))
		tmp = Float64(Float64(y - z) * x);
	else
		tmp = Float64(z + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -24000000000.0) || ~((x <= 1.0)))
		tmp = (y - z) * x;
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -24000000000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision], N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e10 or 1 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. 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. Taylor expanded in x around inf 98.1%

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

   if -2.4e10 < x < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. 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. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in y around inf 99.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -24000000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 5: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+50}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-75}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+50) (* y x) (if (<= y 5.2e-75) (* z (- 1.0 x)) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+50) {
		tmp = y * x;
	} else if (y <= 5.2e-75) {
		tmp = z * (1.0 - x);
	} else {
		tmp = y * x;
	}
	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 <= (-1.7d+50)) then
        tmp = y * x
    else if (y <= 5.2d-75) then
        tmp = z * (1.0d0 - x)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+50) {
		tmp = y * x;
	} else if (y <= 5.2e-75) {
		tmp = z * (1.0 - x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e+50:
		tmp = y * x
	elif y <= 5.2e-75:
		tmp = z * (1.0 - x)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+50)
		tmp = Float64(y * x);
	elseif (y <= 5.2e-75)
		tmp = Float64(z * Float64(1.0 - x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.7e+50)
		tmp = y * x;
	elseif (y <= 5.2e-75)
		tmp = z * (1.0 - x);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+50], N[(y * x), $MachinePrecision], If[LessEqual[y, 5.2e-75], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6999999999999999e50 or 5.2e-75 < y

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 71.6%

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

   if -1.6999999999999999e50 < y < 5.2e-75

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around 0 88.3%

      \[\leadsto \color{blue}{z \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+50}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-75}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 60.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-62}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6e-27) (* y x) (if (<= x 3e-62) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e-27) {
		tmp = y * x;
	} else if (x <= 3e-62) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	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 <= (-6d-27)) then
        tmp = y * x
    else if (x <= 3d-62) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e-27) {
		tmp = y * x;
	} else if (x <= 3e-62) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6e-27:
		tmp = y * x
	elif x <= 3e-62:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6e-27)
		tmp = Float64(y * x);
	elseif (x <= 3e-62)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6e-27)
		tmp = y * x;
	elseif (x <= 3e-62)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6e-27], N[(y * x), $MachinePrecision], If[LessEqual[x, 3e-62], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.0000000000000002e-27 or 3.0000000000000001e-62 < x

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 55.0%

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

   if -6.0000000000000002e-27 < x < 3.0000000000000001e-62

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in x around 0 80.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-62}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7: 35.8% 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%

   \[x \cdot y + \left(1 - x\right) \cdot z \]

2. Taylor expanded in x around 0 40.9%

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

3. Final simplification 40.9%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))