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

Percentage Accurate: 97.6% → 100.0%

Time: 3.7s

Alternatives: 7

Speedup: 1.3×

• 21.5% 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: 97.6% 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 98.0%

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

   1. sub-neg 98.0%

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

   2. +-commutative 98.0%

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

   3. distribute-lft1-in 98.0%

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

   4. associate-+r+ 98.0%

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

   5. +-commutative 98.0%

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

   6. *-commutative 98.0%

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

   7. neg-mul-1 98.0%

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

   8. associate-*r* 98.0%

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

   9. *-commutative 98.0%

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

   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: 61.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-65}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+71} \lor \neg \left(x \leq 1.22 \cdot 10^{+182}\right) \land x \leq 1.85 \cdot 10^{+205}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -4.3e+43)
     t_0
     (if (<= x -1.15e-24)
       (* y x)
       (if (<= x 1.15e-65)
         z
         (if (<= x 1.8e-36)
           (* y x)
           (if (<= x 2.15e-11)
             z
             (if (or (<= x 3.5e+71)
                     (and (not (<= x 1.22e+182)) (<= x 1.85e+205)))
               (* y x)
               t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -4.3e+43) {
		tmp = t_0;
	} else if (x <= -1.15e-24) {
		tmp = y * x;
	} else if (x <= 1.15e-65) {
		tmp = z;
	} else if (x <= 1.8e-36) {
		tmp = y * x;
	} else if (x <= 2.15e-11) {
		tmp = z;
	} else if ((x <= 3.5e+71) || (!(x <= 1.22e+182) && (x <= 1.85e+205))) {
		tmp = y * 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 = z * -x
    if (x <= (-4.3d+43)) then
        tmp = t_0
    else if (x <= (-1.15d-24)) then
        tmp = y * x
    else if (x <= 1.15d-65) then
        tmp = z
    else if (x <= 1.8d-36) then
        tmp = y * x
    else if (x <= 2.15d-11) then
        tmp = z
    else if ((x <= 3.5d+71) .or. (.not. (x <= 1.22d+182)) .and. (x <= 1.85d+205)) then
        tmp = y * 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 = z * -x;
	double tmp;
	if (x <= -4.3e+43) {
		tmp = t_0;
	} else if (x <= -1.15e-24) {
		tmp = y * x;
	} else if (x <= 1.15e-65) {
		tmp = z;
	} else if (x <= 1.8e-36) {
		tmp = y * x;
	} else if (x <= 2.15e-11) {
		tmp = z;
	} else if ((x <= 3.5e+71) || (!(x <= 1.22e+182) && (x <= 1.85e+205))) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -4.3e+43:
		tmp = t_0
	elif x <= -1.15e-24:
		tmp = y * x
	elif x <= 1.15e-65:
		tmp = z
	elif x <= 1.8e-36:
		tmp = y * x
	elif x <= 2.15e-11:
		tmp = z
	elif (x <= 3.5e+71) or (not (x <= 1.22e+182) and (x <= 1.85e+205)):
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -4.3e+43)
		tmp = t_0;
	elseif (x <= -1.15e-24)
		tmp = Float64(y * x);
	elseif (x <= 1.15e-65)
		tmp = z;
	elseif (x <= 1.8e-36)
		tmp = Float64(y * x);
	elseif (x <= 2.15e-11)
		tmp = z;
	elseif ((x <= 3.5e+71) || (!(x <= 1.22e+182) && (x <= 1.85e+205)))
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (x <= -4.3e+43)
		tmp = t_0;
	elseif (x <= -1.15e-24)
		tmp = y * x;
	elseif (x <= 1.15e-65)
		tmp = z;
	elseif (x <= 1.8e-36)
		tmp = y * x;
	elseif (x <= 2.15e-11)
		tmp = z;
	elseif ((x <= 3.5e+71) || (~((x <= 1.22e+182)) && (x <= 1.85e+205)))
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[x, -4.3e+43], t$95$0, If[LessEqual[x, -1.15e-24], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.15e-65], z, If[LessEqual[x, 1.8e-36], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.15e-11], z, If[Or[LessEqual[x, 3.5e+71], And[N[Not[LessEqual[x, 1.22e+182]], $MachinePrecision], LessEqual[x, 1.85e+205]]], N[(y * x), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e43 or 3.4999999999999999e71 < x < 1.22e182 or 1.8499999999999999e205 < x

   1. Initial program 95.5%

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

      1. sub-neg 95.5%

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

      2. +-commutative 95.5%

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

      3. distribute-lft1-in 95.5%

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

      4. associate-+r+ 95.5%

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

      5. +-commutative 95.5%

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

      6. *-commutative 95.5%

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

      7. neg-mul-1 95.5%

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

      8. associate-*r* 95.5%

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

      9. *-commutative 95.5%

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

      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 100.0%

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

   5. Taylor expanded in y around 0 67.1%

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

      1. mul-1-neg 67.1%

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

      2. distribute-rgt-neg-in 67.1%

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

   7. Simplified 67.1%

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

   if -4.3e43 < x < -1.1500000000000001e-24 or 1.15e-65 < x < 1.80000000000000016e-36 or 2.15000000000000001e-11 < x < 3.4999999999999999e71 or 1.22e182 < x < 1.8499999999999999e205

   1. Initial program 97.8%

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

   2. Taylor expanded in y around inf 72.6%

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

   if -1.1500000000000001e-24 < x < 1.15e-65 or 1.80000000000000016e-36 < x < 2.15000000000000001e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+43}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-65}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+71} \lor \neg \left(x \leq 1.22 \cdot 10^{+182}\right) \land x \leq 1.85 \cdot 10^{+205}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]

Alternative 3: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-12}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) x)))
   (if (<= x -1.3e-24)
     t_0
     (if (<= x 4.8e-61)
       (* z (- 1.0 x))
       (if (<= x 7.6e-36) (* y x) (if (<= x 7.1e-12) z t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -1.3e-24) {
		tmp = t_0;
	} else if (x <= 4.8e-61) {
		tmp = z * (1.0 - x);
	} else if (x <= 7.6e-36) {
		tmp = y * x;
	} else if (x <= 7.1e-12) {
		tmp = z;
	} 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 <= (-1.3d-24)) then
        tmp = t_0
    else if (x <= 4.8d-61) then
        tmp = z * (1.0d0 - x)
    else if (x <= 7.6d-36) then
        tmp = y * x
    else if (x <= 7.1d-12) then
        tmp = z
    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 <= -1.3e-24) {
		tmp = t_0;
	} else if (x <= 4.8e-61) {
		tmp = z * (1.0 - x);
	} else if (x <= 7.6e-36) {
		tmp = y * x;
	} else if (x <= 7.1e-12) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) * x
	tmp = 0
	if x <= -1.3e-24:
		tmp = t_0
	elif x <= 4.8e-61:
		tmp = z * (1.0 - x)
	elif x <= 7.6e-36:
		tmp = y * x
	elif x <= 7.1e-12:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * x)
	tmp = 0.0
	if (x <= -1.3e-24)
		tmp = t_0;
	elseif (x <= 4.8e-61)
		tmp = Float64(z * Float64(1.0 - x));
	elseif (x <= 7.6e-36)
		tmp = Float64(y * x);
	elseif (x <= 7.1e-12)
		tmp = z;
	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 <= -1.3e-24)
		tmp = t_0;
	elseif (x <= 4.8e-61)
		tmp = z * (1.0 - x);
	elseif (x <= 7.6e-36)
		tmp = y * x;
	elseif (x <= 7.1e-12)
		tmp = z;
	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, -1.3e-24], t$95$0, If[LessEqual[x, 4.8e-61], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.6e-36], N[(y * x), $MachinePrecision], If[LessEqual[x, 7.1e-12], z, t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3e-24 or 7.1e-12 < x

   1. Initial program 96.1%

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

      1. sub-neg 96.1%

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

      2. +-commutative 96.1%

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

      3. distribute-lft1-in 96.1%

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

      4. associate-+r+ 96.1%

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

      5. +-commutative 96.1%

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

      6. *-commutative 96.1%

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

      7. neg-mul-1 96.1%

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

      8. associate-*r* 96.1%

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

      9. *-commutative 96.1%

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

      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 99.0%

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

   if -1.3e-24 < x < 4.8000000000000002e-61

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.4%

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

   if 4.8000000000000002e-61 < x < 7.59999999999999942e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.4%

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

   if 7.59999999999999942e-36 < x < 7.1e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-12}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \end{array} \]

Alternative 4: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e-25)
   (* y x)
   (if (<= x 6.5e-61)
     z
     (if (<= x 1.8e-36) (* y x) (if (<= x 2.6e-18) z (* y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e-25) {
		tmp = y * x;
	} else if (x <= 6.5e-61) {
		tmp = z;
	} else if (x <= 1.8e-36) {
		tmp = y * x;
	} else if (x <= 2.6e-18) {
		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 <= (-3.6d-25)) then
        tmp = y * x
    else if (x <= 6.5d-61) then
        tmp = z
    else if (x <= 1.8d-36) then
        tmp = y * x
    else if (x <= 2.6d-18) 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 <= -3.6e-25) {
		tmp = y * x;
	} else if (x <= 6.5e-61) {
		tmp = z;
	} else if (x <= 1.8e-36) {
		tmp = y * x;
	} else if (x <= 2.6e-18) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e-25:
		tmp = y * x
	elif x <= 6.5e-61:
		tmp = z
	elif x <= 1.8e-36:
		tmp = y * x
	elif x <= 2.6e-18:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e-25)
		tmp = Float64(y * x);
	elseif (x <= 6.5e-61)
		tmp = z;
	elseif (x <= 1.8e-36)
		tmp = Float64(y * x);
	elseif (x <= 2.6e-18)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.6e-25)
		tmp = y * x;
	elseif (x <= 6.5e-61)
		tmp = z;
	elseif (x <= 1.8e-36)
		tmp = y * x;
	elseif (x <= 2.6e-18)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e-25], N[(y * x), $MachinePrecision], If[LessEqual[x, 6.5e-61], z, If[LessEqual[x, 1.8e-36], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.6e-18], z, N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999999e-25 or 6.4999999999999994e-61 < x < 1.80000000000000016e-36 or 2.6e-18 < x

   1. Initial program 96.3%

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

   2. Taylor expanded in y around inf 51.0%

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

   if -3.5999999999999999e-25 < x < 6.4999999999999994e-61 or 1.80000000000000016e-36 < x < 2.6e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-182} \lor \neg \left(z \leq 1.2 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7e-182) (not (<= z 1.2e-58))) (* z (- 1.0 x)) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7e-182) || !(z <= 1.2e-58)) {
		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 ((z <= (-7d-182)) .or. (.not. (z <= 1.2d-58))) 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 ((z <= -7e-182) || !(z <= 1.2e-58)) {
		tmp = z * (1.0 - x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7e-182) or not (z <= 1.2e-58):
		tmp = z * (1.0 - x)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7e-182) || !(z <= 1.2e-58))
		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 ((z <= -7e-182) || ~((z <= 1.2e-58)))
		tmp = z * (1.0 - x);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7e-182], N[Not[LessEqual[z, 1.2e-58]], $MachinePrecision]], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999966e-182 or 1.2e-58 < z

   1. Initial program 97.1%

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

   2. Taylor expanded in y around 0 83.1%

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

   if -6.99999999999999966e-182 < z < 1.2e-58

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 74.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-182} \lor \neg \left(z \leq 1.2 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{+16} \lor \neg \left(x \leq 0.43\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.35e+16) (not (<= x 0.43))) (* (- y z) x) (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.35e+16) || !(x <= 0.43)) {
		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 <= (-3.35d+16)) .or. (.not. (x <= 0.43d0))) 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 <= -3.35e+16) || !(x <= 0.43)) {
		tmp = (y - z) * x;
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.35e+16) or not (x <= 0.43):
		tmp = (y - z) * x
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.35e+16) || !(x <= 0.43))
		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 <= -3.35e+16) || ~((x <= 0.43)))
		tmp = (y - z) * x;
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.35e+16], N[Not[LessEqual[x, 0.43]], $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 < -3.35e16 or 0.429999999999999993 < x

   1. Initial program 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. distribute-lft1-in 95.9%

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

      4. associate-+r+ 95.9%

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

      5. +-commutative 95.9%

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

      6. *-commutative 95.9%

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

      7. neg-mul-1 95.9%

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

      8. associate-*r* 95.9%

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

      9. *-commutative 95.9%

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

      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 99.7%

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

   if -3.35e16 < x < 0.429999999999999993

   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{+16} \lor \neg \left(x \leq 0.43\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 7: 36.2% 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 98.0%

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

2. Taylor expanded in x around 0 38.9%

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

3. Final simplification 38.9%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))