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

Percentage Accurate: 98.0% → 100.0%

Time: 3.6s

Alternatives: 8

Speedup: 1.3×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((1.0d0 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((1.0d0 - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. distribute-lft-out-- 98.4%

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

   3. *-rgt-identity 98.4%

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

   4. cancel-sign-sub-inv 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-+r+ 98.4%

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

   7. *-commutative 98.4%

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

   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: 61.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+190}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1250000000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-78}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.95e+190)
   (* x y)
   (if (<= x -1250000000.0)
     (* x (- z))
     (if (<= x -3.4e-78) (* x y) (if (<= x 3.6e-14) z (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.95e+190) {
		tmp = x * y;
	} else if (x <= -1250000000.0) {
		tmp = x * -z;
	} else if (x <= -3.4e-78) {
		tmp = x * y;
	} else if (x <= 3.6e-14) {
		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 <= (-2.95d+190)) then
        tmp = x * y
    else if (x <= (-1250000000.0d0)) then
        tmp = x * -z
    else if (x <= (-3.4d-78)) then
        tmp = x * y
    else if (x <= 3.6d-14) 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 <= -2.95e+190) {
		tmp = x * y;
	} else if (x <= -1250000000.0) {
		tmp = x * -z;
	} else if (x <= -3.4e-78) {
		tmp = x * y;
	} else if (x <= 3.6e-14) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.95e+190:
		tmp = x * y
	elif x <= -1250000000.0:
		tmp = x * -z
	elif x <= -3.4e-78:
		tmp = x * y
	elif x <= 3.6e-14:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.95e+190)
		tmp = Float64(x * y);
	elseif (x <= -1250000000.0)
		tmp = Float64(x * Float64(-z));
	elseif (x <= -3.4e-78)
		tmp = Float64(x * y);
	elseif (x <= 3.6e-14)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.95e+190)
		tmp = x * y;
	elseif (x <= -1250000000.0)
		tmp = x * -z;
	elseif (x <= -3.4e-78)
		tmp = x * y;
	elseif (x <= 3.6e-14)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.95e+190], N[(x * y), $MachinePrecision], If[LessEqual[x, -1250000000.0], N[(x * (-z)), $MachinePrecision], If[LessEqual[x, -3.4e-78], N[(x * y), $MachinePrecision], If[LessEqual[x, 3.6e-14], z, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.94999999999999986e190 or -1.25e9 < x < -3.40000000000000012e-78 or 3.5999999999999998e-14 < x

   1. Initial program 96.0%

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

   2. Taylor expanded in y around inf 66.4%

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

   if -2.94999999999999986e190 < x < -1.25e9

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 67.4%

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

   3. Taylor expanded in x around inf 65.1%

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

      1. associate-*r* 65.1%

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

      2. mul-1-neg 65.1%

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

   5. Simplified 65.1%

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

   if -3.40000000000000012e-78 < x < 3.5999999999999998e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+190}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1250000000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-78}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-75} \lor \neg \left(x \leq 0.0005\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.65e-75) (not (<= x 0.0005))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e-75) || !(x <= 0.0005)) {
		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 <= (-1.65d-75)) .or. (.not. (x <= 0.0005d0))) 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 <= -1.65e-75) || !(x <= 0.0005)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.65e-75) or not (x <= 0.0005):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.65e-75) || !(x <= 0.0005))
		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 <= -1.65e-75) || ~((x <= 0.0005)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.65e-75], N[Not[LessEqual[x, 0.0005]], $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 < -1.65e-75 or 5.0000000000000001e-4 < x

   1. Initial program 97.1%

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

   2. Taylor expanded in x around inf 96.2%

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

      1. neg-mul-1 96.2%

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

      2. +-commutative 96.2%

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

      3. unsub-neg 96.2%

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

   4. Simplified 96.2%

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

   if -1.65e-75 < x < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-75} \lor \neg \left(x \leq 0.0005\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45000 \lor \neg \left(x \leq 0.057\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -45000.0) (not (<= x 0.057))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -45000.0) || !(x <= 0.057)) {
		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 <= (-45000.0d0)) .or. (.not. (x <= 0.057d0))) 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 <= -45000.0) || !(x <= 0.057)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -45000.0) or not (x <= 0.057):
		tmp = x * (y - z)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -45000.0) || !(x <= 0.057))
		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 <= -45000.0) || ~((x <= 0.057)))
		tmp = x * (y - z);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -45000.0], N[Not[LessEqual[x, 0.057]], $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 < -45000 or 0.0570000000000000021 < x

   1. Initial program 96.8%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. +-commutative 98.7%

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

      3. unsub-neg 98.7%

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

   4. Simplified 98.7%

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

   if -45000 < x < 0.0570000000000000021

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. unsub-neg 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -45000 \lor \neg \left(x \leq 0.057\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 5: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+38}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e+38) (* x y) (if (<= y 1.65e-39) (* z (- 1.0 x)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e+38) {
		tmp = x * y;
	} else if (y <= 1.65e-39) {
		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 <= (-1.15d+38)) then
        tmp = x * y
    else if (y <= 1.65d-39) 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 <= -1.15e+38) {
		tmp = x * y;
	} else if (y <= 1.65e-39) {
		tmp = z * (1.0 - x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.15e+38:
		tmp = x * y
	elif y <= 1.65e-39:
		tmp = z * (1.0 - x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.15e+38)
		tmp = Float64(x * y);
	elseif (y <= 1.65e-39)
		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 <= -1.15e+38)
		tmp = x * y;
	elseif (y <= 1.65e-39)
		tmp = z * (1.0 - x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.15e+38], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.65e-39], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1500000000000001e38 or 1.64999999999999992e-39 < y

   1. Initial program 96.8%

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

   2. Taylor expanded in y around inf 69.3%

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

   if -1.1500000000000001e38 < y < 1.64999999999999992e-39

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 86.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+38}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-71}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-71) (* x y) (if (<= x 1.15e-14) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-71) {
		tmp = x * y;
	} else if (x <= 1.15e-14) {
		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 <= (-2.3d-71)) then
        tmp = x * y
    else if (x <= 1.15d-14) 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 <= -2.3e-71) {
		tmp = x * y;
	} else if (x <= 1.15e-14) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.3e-71:
		tmp = x * y
	elif x <= 1.15e-14:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-71)
		tmp = Float64(x * y);
	elseif (x <= 1.15e-14)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e-71)
		tmp = x * y;
	elseif (x <= 1.15e-14)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e-71], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.15e-14], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998e-71 or 1.14999999999999999e-14 < x

   1. Initial program 97.1%

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

   2. Taylor expanded in y around inf 57.9%

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

   if -2.2999999999999998e-71 < x < 1.14999999999999999e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-71}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

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 98.4%

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

2. Taylor expanded in x around 0 100.0%

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

3. Taylor expanded in z around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. neg-mul-1 98.4%

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

   3. distribute-lft-in 98.4%

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

   4. *-rgt-identity 98.4%

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

   5. associate-+l+ 98.4%

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

   6. +-commutative 98.4%

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

   7. distribute-rgt-neg-out 98.4%

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

   8. unsub-neg 98.4%

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

   9. distribute-rgt-out-- 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 8: 36.9% 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 98.4%

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

2. Taylor expanded in x around 0 38.8%

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

3. Final simplification 38.8%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))