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

Percentage Accurate: 97.6% → 100.0%

Time: 8.3s

Alternatives: 8

Speedup: 1.3×

• 20.9% 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, 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 99.6%

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

   1. +-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-lft-in 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-+l+ 99.6%

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

   7. *-lft-identity 99.6%

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

   8. +-commutative 99.6%

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

   9. *-commutative 99.6%

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

   10. distribute-lft-neg-out 99.6%

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

   11. distribute-rgt-neg-out 99.6%

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

   12. distribute-lft-out 100.0%

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

   13. fma-def 100.0%

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

   14. +-commutative 100.0%

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

   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. Final simplification 100.0%

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

Alternative 2: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+143} \lor \neg \left(x \leq 4.2 \cdot 10^{+260}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -19.0)
   (* x y)
   (if (<= x 1.0)
     z
     (if (or (<= x 1.2e+143) (not (<= x 4.2e+260))) (* x (- z)) (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -19.0) {
		tmp = x * y;
	} else if (x <= 1.0) {
		tmp = z;
	} else if ((x <= 1.2e+143) || !(x <= 4.2e+260)) {
		tmp = x * -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 <= (-19.0d0)) then
        tmp = x * y
    else if (x <= 1.0d0) then
        tmp = z
    else if ((x <= 1.2d+143) .or. (.not. (x <= 4.2d+260))) then
        tmp = x * -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 <= -19.0) {
		tmp = x * y;
	} else if (x <= 1.0) {
		tmp = z;
	} else if ((x <= 1.2e+143) || !(x <= 4.2e+260)) {
		tmp = x * -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -19.0:
		tmp = x * y
	elif x <= 1.0:
		tmp = z
	elif (x <= 1.2e+143) or not (x <= 4.2e+260):
		tmp = x * -z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(x * y);
	elseif (x <= 1.0)
		tmp = z;
	elseif ((x <= 1.2e+143) || !(x <= 4.2e+260))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -19.0)
		tmp = x * y;
	elseif (x <= 1.0)
		tmp = z;
	elseif ((x <= 1.2e+143) || ~((x <= 4.2e+260)))
		tmp = x * -z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -19.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.0], z, If[Or[LessEqual[x, 1.2e+143], N[Not[LessEqual[x, 4.2e+260]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -19 or 1.1999999999999999e143 < x < 4.20000000000000025e260

   1. Initial program 98.8%

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

   2. Taylor expanded in y around inf 58.2%

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

   if -19 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 76.2%

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

   if 1 < x < 1.1999999999999999e143 or 4.20000000000000025e260 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. sub-neg 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in y around 0 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

      3. mul-1-neg 63.6%

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

   7. Simplified 63.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+143} \lor \neg \left(x \leq 4.2 \cdot 10^{+260}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-7} \lor \neg \left(x \leq 0.0075\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e-7) (not (<= x 0.0075))) (* x (- y z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-7) || !(x <= 0.0075)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.9d-7)) .or. (.not. (x <= 0.0075d0))) then
        tmp = x * (y - z)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-7) || !(x <= 0.0075)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e-7) or not (x <= 0.0075):
		tmp = x * (y - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e-7) || !(x <= 0.0075))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.9e-7) || ~((x <= 0.0075)))
		tmp = x * (y - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e-7], N[Not[LessEqual[x, 0.0075]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999998e-7 or 0.0074999999999999997 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. sub-neg 97.8%

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

   4. Simplified 97.8%

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

   if -2.8999999999999998e-7 < x < 0.0074999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 76.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-7} \lor \neg \left(x \leq 0.0075\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -62 \lor \neg \left(x \leq 1800000000000\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -62.0) (not (<= x 1800000000000.0)))
   (* x (- y z))
   (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -62.0) || !(x <= 1800000000000.0)) {
		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 <= (-62.0d0)) .or. (.not. (x <= 1800000000000.0d0))) 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 <= -62.0) || !(x <= 1800000000000.0)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -62.0) or not (x <= 1800000000000.0):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -62.0) || !(x <= 1800000000000.0))
		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 <= -62.0) || ~((x <= 1800000000000.0)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -62.0], N[Not[LessEqual[x, 1800000000000.0]], $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 < -62 or 1.8e12 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

   4. Simplified 99.5%

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

   if -62 < x < 1.8e12

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62 \lor \neg \left(x \leq 1800000000000\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		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 <= 1.0d0))) 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 <= 1.0)) {
		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 <= 1.0):
		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 <= 1.0))
		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 <= 1.0)))
		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, 1.0]], $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 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. sub-neg 97.8%

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

   4. Simplified 97.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-+l+ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. distribute-lft-out 100.0%

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

      13. fma-def 100.0%

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

      14. +-commutative 100.0%

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

      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. Step-by-step derivation

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. distribute-rgt-neg-out 100.0%

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

      6. flip3-+ 46.2%

         \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{3} + {\left(z \cdot \left(-x\right)\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot \left(-x\right)\right) \cdot \left(z \cdot \left(-x\right)\right) - \left(x \cdot y\right) \cdot \left(z \cdot \left(-x\right)\right)\right)}} + z \]

      7. clear-num 46.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot \left(-x\right)\right) \cdot \left(z \cdot \left(-x\right)\right) - \left(x \cdot y\right) \cdot \left(z \cdot \left(-x\right)\right)\right)}{{\left(x \cdot y\right)}^{3} + {\left(z \cdot \left(-x\right)\right)}^{3}}}} + z \]

      8. *-un-lft-identity 46.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot \left(-x\right)\right) \cdot \left(z \cdot \left(-x\right)\right) - \left(x \cdot y\right) \cdot \left(z \cdot \left(-x\right)\right)\right)\right)}}{{\left(x \cdot y\right)}^{3} + {\left(z \cdot \left(-x\right)\right)}^{3}}} + z \]

      9. associate-/l* 46.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot y\right)}^{3} + {\left(z \cdot \left(-x\right)\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(z \cdot \left(-x\right)\right) \cdot \left(z \cdot \left(-x\right)\right) - \left(x \cdot y\right) \cdot \left(z \cdot \left(-x\right)\right)\right)}}}} + z \]

      10. flip3-+ 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. distribute-lft-neg-in 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x}}{y - z}}} + z \]

   8. Taylor expanded in y around inf 99.1%

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

4. Final simplification 98.4%

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

Alternative 6: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19 \lor \neg \left(x \leq 0.0075\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -19.0) (not (<= x 0.0075))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -19.0) || !(x <= 0.0075)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-19.0d0)) .or. (.not. (x <= 0.0075d0))) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -19.0) || !(x <= 0.0075)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -19.0) or not (x <= 0.0075):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -19.0) || !(x <= 0.0075))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -19.0) || ~((x <= 0.0075)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -19.0], N[Not[LessEqual[x, 0.0075]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -19 or 0.0074999999999999997 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 51.2%

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

   if -19 < x < 0.0074999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 76.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19 \lor \neg \left(x \leq 0.0075\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

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 99.6%

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

   1. +-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-lft-in 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-+l+ 99.6%

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

   7. *-lft-identity 99.6%

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

   8. +-commutative 99.6%

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

   9. *-commutative 99.6%

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

   10. distribute-lft-neg-out 99.6%

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

   11. distribute-rgt-neg-out 99.6%

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

   12. distribute-lft-out 100.0%

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

   13. fma-def 100.0%

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

   14. +-commutative 100.0%

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

   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. Step-by-step derivation

   1. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 8: 36.4% 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 99.6%

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

2. Taylor expanded in x around 0 39.0%

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

3. Final simplification 39.0%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))