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

Percentage Accurate: 97.9% → 100.0%

Time: 4.2s

Alternatives: 8

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: 97.9% 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 98.0%

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

   1. *-commutative 98.0%

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

   2. distribute-lft-out-- 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. cancel-sign-sub-inv 98.0%

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

   5. +-commutative 98.0%

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

   6. associate-+r+ 98.0%

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

   7. *-commutative 98.0%

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

   8. distribute-rgt-out 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-def 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. Final simplification 100.0%

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

Alternative 2: 61.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -165000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-12}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+152}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+269}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= x -165000000000.0)
     t_0
     (if (<= x -4.2e-79)
       (* x y)
       (if (<= x 1.6e-12)
         z
         (if (<= x 2.1e+152) (* x y) (if (<= x 7e+269) t_0 (* x y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (x <= -165000000000.0) {
		tmp = t_0;
	} else if (x <= -4.2e-79) {
		tmp = x * y;
	} else if (x <= 1.6e-12) {
		tmp = z;
	} else if (x <= 2.1e+152) {
		tmp = x * y;
	} else if (x <= 7e+269) {
		tmp = t_0;
	} 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 = x * -z
    if (x <= (-165000000000.0d0)) then
        tmp = t_0
    else if (x <= (-4.2d-79)) then
        tmp = x * y
    else if (x <= 1.6d-12) then
        tmp = z
    else if (x <= 2.1d+152) then
        tmp = x * y
    else if (x <= 7d+269) then
        tmp = t_0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (x <= -165000000000.0) {
		tmp = t_0;
	} else if (x <= -4.2e-79) {
		tmp = x * y;
	} else if (x <= 1.6e-12) {
		tmp = z;
	} else if (x <= 2.1e+152) {
		tmp = x * y;
	} else if (x <= 7e+269) {
		tmp = t_0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if x <= -165000000000.0:
		tmp = t_0
	elif x <= -4.2e-79:
		tmp = x * y
	elif x <= 1.6e-12:
		tmp = z
	elif x <= 2.1e+152:
		tmp = x * y
	elif x <= 7e+269:
		tmp = t_0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (x <= -165000000000.0)
		tmp = t_0;
	elseif (x <= -4.2e-79)
		tmp = Float64(x * y);
	elseif (x <= 1.6e-12)
		tmp = z;
	elseif (x <= 2.1e+152)
		tmp = Float64(x * y);
	elseif (x <= 7e+269)
		tmp = t_0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (x <= -165000000000.0)
		tmp = t_0;
	elseif (x <= -4.2e-79)
		tmp = x * y;
	elseif (x <= 1.6e-12)
		tmp = z;
	elseif (x <= 2.1e+152)
		tmp = x * y;
	elseif (x <= 7e+269)
		tmp = t_0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[x, -165000000000.0], t$95$0, If[LessEqual[x, -4.2e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.6e-12], z, If[LessEqual[x, 2.1e+152], N[(x * y), $MachinePrecision], If[LessEqual[x, 7e+269], t$95$0, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e11 or 2.1000000000000002e152 < x < 7.0000000000000003e269

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. distribute-lft-out-- 96.3%

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

      3. *-rgt-identity 96.3%

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

      4. cancel-sign-sub-inv 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+r+ 96.3%

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

      7. *-commutative 96.3%

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

      8. distribute-rgt-out 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.6%

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

   5. Taylor expanded in y around 0 70.9%

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

      1. mul-1-neg 70.9%

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

      2. distribute-rgt-neg-in 70.9%

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

   7. Simplified 70.9%

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

   if -1.65e11 < x < -4.1999999999999999e-79 or 1.6e-12 < x < 2.1000000000000002e152 or 7.0000000000000003e269 < x

   1. Initial program 97.0%

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

   2. Taylor expanded in y around inf 70.4%

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

   if -4.1999999999999999e-79 < x < 1.6e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -165000000000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-12}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+152}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+269}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-40} \lor \neg \left(y \leq 4.6 \cdot 10^{-6}\right) \land y \leq 2.45 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.05e+72)
   (* x y)
   (if (or (<= y 4.5e-40) (and (not (<= y 4.6e-6)) (<= y 2.45e+113)))
     (* z (- 1.0 x))
     (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.05e+72) {
		tmp = x * y;
	} else if ((y <= 4.5e-40) || (!(y <= 4.6e-6) && (y <= 2.45e+113))) {
		tmp = z * (1.0 - x);
	} 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 <= (-3.05d+72)) then
        tmp = x * y
    else if ((y <= 4.5d-40) .or. (.not. (y <= 4.6d-6)) .and. (y <= 2.45d+113)) then
        tmp = z * (1.0d0 - x)
    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 <= -3.05e+72) {
		tmp = x * y;
	} else if ((y <= 4.5e-40) || (!(y <= 4.6e-6) && (y <= 2.45e+113))) {
		tmp = z * (1.0 - x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.05e+72:
		tmp = x * y
	elif (y <= 4.5e-40) or (not (y <= 4.6e-6) and (y <= 2.45e+113)):
		tmp = z * (1.0 - x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.05e+72)
		tmp = Float64(x * y);
	elseif ((y <= 4.5e-40) || (!(y <= 4.6e-6) && (y <= 2.45e+113)))
		tmp = Float64(z * Float64(1.0 - x));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.05e+72)
		tmp = x * y;
	elseif ((y <= 4.5e-40) || (~((y <= 4.6e-6)) && (y <= 2.45e+113)))
		tmp = z * (1.0 - x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.05e+72], N[(x * y), $MachinePrecision], If[Or[LessEqual[y, 4.5e-40], And[N[Not[LessEqual[y, 4.6e-6]], $MachinePrecision], LessEqual[y, 2.45e+113]]], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.04999999999999996e72 or 4.5000000000000001e-40 < y < 4.6e-6 or 2.45000000000000011e113 < y

   1. Initial program 95.8%

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

   2. Taylor expanded in y around inf 78.5%

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

   if -3.04999999999999996e72 < y < 4.5000000000000001e-40 or 4.6e-6 < y < 2.45000000000000011e113

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 87.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-40} \lor \neg \left(y \leq 4.6 \cdot 10^{-6}\right) \land y \leq 2.45 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-79} \lor \neg \left(x \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.2e-79) (not (<= x 1.2e-7))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.2e-79) || !(x <= 1.2e-7)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - 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 <= (-4.2d-79)) .or. (.not. (x <= 1.2d-7))) then
        tmp = x * (y - z)
    else
        tmp = z * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.2e-79) || !(x <= 1.2e-7)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.2e-79) or not (x <= 1.2e-7):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.2e-79) || !(x <= 1.2e-7))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.2e-79) || ~((x <= 1.2e-7)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.2e-79], N[Not[LessEqual[x, 1.2e-7]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999999e-79 or 1.19999999999999989e-7 < x

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. distribute-lft-out-- 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. cancel-sign-sub-inv 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-+r+ 96.5%

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

      7. *-commutative 96.5%

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

      8. distribute-rgt-out 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 95.9%

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

   if -4.1999999999999999e-79 < x < 1.19999999999999989e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-79} \lor \neg \left(x \leq 1.2 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.4e-7))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.4e-7)) {
		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 <= 1.4d-7))) 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 <= 1.4e-7)) {
		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 <= 1.4e-7):
		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 <= 1.4e-7))
		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 <= 1.4e-7)))
		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, 1.4e-7]], $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 1.4000000000000001e-7 < x

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-lft-out-- 96.2%

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

      3. *-rgt-identity 96.2%

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

      4. cancel-sign-sub-inv 96.2%

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

      5. +-commutative 96.2%

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

      6. associate-+r+ 96.2%

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

      7. *-commutative 96.2%

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

      8. distribute-rgt-out 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. +-commutative 99.9%

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

      12. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1.4000000000000001e-7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-out-- 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+r+ 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-out 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 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. 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 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 6: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e-79) (* x y) (if (<= x 8.6e-15) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e-79) {
		tmp = x * y;
	} else if (x <= 8.6e-15) {
		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 (x <= (-4.2d-79)) then
        tmp = x * y
    else if (x <= 8.6d-15) 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 (x <= -4.2e-79) {
		tmp = x * y;
	} else if (x <= 8.6e-15) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.2e-79:
		tmp = x * y
	elif x <= 8.6e-15:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.2e-79)
		tmp = Float64(x * y);
	elseif (x <= 8.6e-15)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.2e-79)
		tmp = x * y;
	elseif (x <= 8.6e-15)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.2e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 8.6e-15], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999999e-79 or 8.5999999999999993e-15 < x

   1. Initial program 96.6%

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

   2. Taylor expanded in y around inf 49.5%

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

   if -4.1999999999999999e-79 < x < 8.5999999999999993e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.3%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 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 98.0%

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

   1. *-commutative 98.0%

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

   2. distribute-lft-out-- 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. cancel-sign-sub-inv 98.0%

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

   5. +-commutative 98.0%

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

   6. associate-+r+ 98.0%

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

   7. *-commutative 98.0%

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

   8. distribute-rgt-out 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-def 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. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 35.9% 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 35.9%

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

3. Final simplification 35.9%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))