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

Percentage Accurate: 98.0% → 100.0%

Time: 4.7s

Alternatives: 6

Speedup: 1.3×

• 20.5% 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: 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, 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 99.2%

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

   1. +-commutative 99.2%

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

   2. remove-double-neg 99.2%

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

   3. distribute-rgt-neg-out 99.2%

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

   4. neg-sub0 99.2%

      \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

   5. neg-sub0 99.2%

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

   6. *-commutative 99.2%

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

   7. distribute-lft-neg-in 99.2%

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

   8. remove-double-neg 99.2%

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

   9. distribute-rgt-out-- 99.2%

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

   10. *-lft-identity 99.2%

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

   11. associate-+l- 99.2%

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

   12. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y - z\right) \]
6. Add Preprocessing

Alternative 2: 61.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+202}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{+77}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.0042:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+100}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -2e+202)
     (* x y)
     (if (<= x -1.5e+166)
       t_0
       (if (<= x -5.4e+77)
         (* x y)
         (if (<= x -2.5e+15)
           t_0
           (if (<= x -4.2e-6)
             (* x y)
             (if (<= x 0.0042) z (if (<= x 3.5e+100) (* x y) t_0)))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -2e+202) {
		tmp = x * y;
	} else if (x <= -1.5e+166) {
		tmp = t_0;
	} else if (x <= -5.4e+77) {
		tmp = x * y;
	} else if (x <= -2.5e+15) {
		tmp = t_0;
	} else if (x <= -4.2e-6) {
		tmp = x * y;
	} else if (x <= 0.0042) {
		tmp = z;
	} else if (x <= 3.5e+100) {
		tmp = x * y;
	} 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 <= (-2d+202)) then
        tmp = x * y
    else if (x <= (-1.5d+166)) then
        tmp = t_0
    else if (x <= (-5.4d+77)) then
        tmp = x * y
    else if (x <= (-2.5d+15)) then
        tmp = t_0
    else if (x <= (-4.2d-6)) then
        tmp = x * y
    else if (x <= 0.0042d0) then
        tmp = z
    else if (x <= 3.5d+100) then
        tmp = x * y
    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 <= -2e+202) {
		tmp = x * y;
	} else if (x <= -1.5e+166) {
		tmp = t_0;
	} else if (x <= -5.4e+77) {
		tmp = x * y;
	} else if (x <= -2.5e+15) {
		tmp = t_0;
	} else if (x <= -4.2e-6) {
		tmp = x * y;
	} else if (x <= 0.0042) {
		tmp = z;
	} else if (x <= 3.5e+100) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -2e+202:
		tmp = x * y
	elif x <= -1.5e+166:
		tmp = t_0
	elif x <= -5.4e+77:
		tmp = x * y
	elif x <= -2.5e+15:
		tmp = t_0
	elif x <= -4.2e-6:
		tmp = x * y
	elif x <= 0.0042:
		tmp = z
	elif x <= 3.5e+100:
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -2e+202)
		tmp = Float64(x * y);
	elseif (x <= -1.5e+166)
		tmp = t_0;
	elseif (x <= -5.4e+77)
		tmp = Float64(x * y);
	elseif (x <= -2.5e+15)
		tmp = t_0;
	elseif (x <= -4.2e-6)
		tmp = Float64(x * y);
	elseif (x <= 0.0042)
		tmp = z;
	elseif (x <= 3.5e+100)
		tmp = Float64(x * y);
	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 <= -2e+202)
		tmp = x * y;
	elseif (x <= -1.5e+166)
		tmp = t_0;
	elseif (x <= -5.4e+77)
		tmp = x * y;
	elseif (x <= -2.5e+15)
		tmp = t_0;
	elseif (x <= -4.2e-6)
		tmp = x * y;
	elseif (x <= 0.0042)
		tmp = z;
	elseif (x <= 3.5e+100)
		tmp = x * y;
	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, -2e+202], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.5e+166], t$95$0, If[LessEqual[x, -5.4e+77], N[(x * y), $MachinePrecision], If[LessEqual[x, -2.5e+15], t$95$0, If[LessEqual[x, -4.2e-6], N[(x * y), $MachinePrecision], If[LessEqual[x, 0.0042], z, If[LessEqual[x, 3.5e+100], N[(x * y), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999998e202 or -1.49999999999999999e166 < x < -5.3999999999999997e77 or -2.5e15 < x < -4.1999999999999996e-6 or 0.00419999999999999974 < x < 3.49999999999999976e100

   1. Initial program 98.2%

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

      1. fma-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in y around inf 70.4%

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

   if -1.9999999999999998e202 < x < -1.49999999999999999e166 or -5.3999999999999997e77 < x < -2.5e15 or 3.49999999999999976e100 < x

   1. Initial program 98.5%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. mul-1-neg 73.0%

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

   10. Simplified 73.0%

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

   if -4.1999999999999996e-6 < x < 0.00419999999999999974

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+202}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+166}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{+77}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.0042:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+100}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-5} \lor \neg \left(x \leq 0.0024\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.02e-5) (not (<= x 0.0024))) (* x (- y z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e-5) || !(x <= 0.0024)) {
		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 <= (-1.02d-5)) .or. (.not. (x <= 0.0024d0))) 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 <= -1.02e-5) || !(x <= 0.0024)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.02e-5) or not (x <= 0.0024):
		tmp = x * (y - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.02e-5) || !(x <= 0.0024))
		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 <= -1.02e-5) || ~((x <= 0.0024)))
		tmp = x * (y - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.02e-5], N[Not[LessEqual[x, 0.0024]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -1.0200000000000001e-5 or 0.00239999999999999979 < x

   1. Initial program 98.4%

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

      1. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

   7. Simplified 99.0%

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

   if -1.0200000000000001e-5 < x < 0.00239999999999999979

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-5} \lor \neg \left(x \leq 0.0024\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -39000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -39000000.0) (not (<= x 1.0))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -39000000.0) || !(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 <= (-39000000.0d0)) .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 <= -39000000.0) || !(x <= 1.0)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -39000000.0) 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 <= -39000000.0) || !(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 <= -39000000.0) || ~((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, -39000000.0], 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 < -3.9e7 or 1 < x

   1. Initial program 98.3%

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

      1. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

   7. Simplified 99.0%

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

   if -3.9e7 < 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. remove-double-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

      5. neg-sub0 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. *-lft-identity 100.0%

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

      11. associate-+l- 100.0%

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

      12. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. distribute-lft-neg-in 99.0%

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

      3. *-commutative 99.0%

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

   7. Simplified 99.0%

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

      1. sub-neg 99.0%

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

      2. +-commutative 99.0%

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

      3. distribute-rgt-neg-out 99.0%

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

      4. remove-double-neg 99.0%

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

   9. Applied egg-rr 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -39000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-7} \lor \neg \left(x \leq 0.0024\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.2e-7) (not (<= x 0.0024))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.2e-7) || !(x <= 0.0024)) {
		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 <= (-2.2d-7)) .or. (.not. (x <= 0.0024d0))) 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 <= -2.2e-7) || !(x <= 0.0024)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.2e-7) or not (x <= 0.0024):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.2e-7) || !(x <= 0.0024))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.2e-7) || ~((x <= 0.0024)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.2e-7], N[Not[LessEqual[x, 0.0024]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000001e-7 or 0.00239999999999999979 < x

   1. Initial program 98.4%

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

      1. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 52.2%

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

   if -2.2000000000000001e-7 < x < 0.00239999999999999979

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-7} \lor \neg \left(x \leq 0.0024\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.3% 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 99.2%

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

   1. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 40.9%

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

6. Final simplification 40.9%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))