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

Percentage Accurate: 97.7% → 100.0%

Time: 3.6s

Alternatives: 8

Speedup: 1.3×

• 21.8% 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: 97.7% 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 99.2%

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

   1. sub-neg 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-lft1-in 99.2%

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

   4. associate-+r+ 99.2%

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

   5. +-commutative 99.2%

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

   6. *-commutative 99.2%

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

   7. neg-mul-1 99.2%

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

   8. associate-*r* 99.2%

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

   9. *-commutative 99.2%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-29}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6200000000:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+206} \lor \neg \left(x \leq 3.9 \cdot 10^{+251}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= x -1.4e+55)
     t_0
     (if (<= x -2.3e-29)
       (* x y)
       (if (<= x 6200000000.0)
         z
         (if (or (<= x 5.1e+206) (not (<= x 3.9e+251))) t_0 (* x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (x <= -1.4e+55) {
		tmp = t_0;
	} else if (x <= -2.3e-29) {
		tmp = x * y;
	} else if (x <= 6200000000.0) {
		tmp = z;
	} else if ((x <= 5.1e+206) || !(x <= 3.9e+251)) {
		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 <= (-1.4d+55)) then
        tmp = t_0
    else if (x <= (-2.3d-29)) then
        tmp = x * y
    else if (x <= 6200000000.0d0) then
        tmp = z
    else if ((x <= 5.1d+206) .or. (.not. (x <= 3.9d+251))) 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 <= -1.4e+55) {
		tmp = t_0;
	} else if (x <= -2.3e-29) {
		tmp = x * y;
	} else if (x <= 6200000000.0) {
		tmp = z;
	} else if ((x <= 5.1e+206) || !(x <= 3.9e+251)) {
		tmp = t_0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if x <= -1.4e+55:
		tmp = t_0
	elif x <= -2.3e-29:
		tmp = x * y
	elif x <= 6200000000.0:
		tmp = z
	elif (x <= 5.1e+206) or not (x <= 3.9e+251):
		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 <= -1.4e+55)
		tmp = t_0;
	elseif (x <= -2.3e-29)
		tmp = Float64(x * y);
	elseif (x <= 6200000000.0)
		tmp = z;
	elseif ((x <= 5.1e+206) || !(x <= 3.9e+251))
		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 <= -1.4e+55)
		tmp = t_0;
	elseif (x <= -2.3e-29)
		tmp = x * y;
	elseif (x <= 6200000000.0)
		tmp = z;
	elseif ((x <= 5.1e+206) || ~((x <= 3.9e+251)))
		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, -1.4e+55], t$95$0, If[LessEqual[x, -2.3e-29], N[(x * y), $MachinePrecision], If[LessEqual[x, 6200000000.0], z, If[Or[LessEqual[x, 5.1e+206], N[Not[LessEqual[x, 3.9e+251]], $MachinePrecision]], t$95$0, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e55 or 6.2e9 < x < 5.1000000000000003e206 or 3.89999999999999976e251 < x

   1. Initial program 98.2%

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

      1. sub-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. distribute-lft1-in 98.2%

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

      4. associate-+r+ 98.2%

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

      5. +-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. neg-mul-1 98.2%

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

      8. associate-*r* 98.2%

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

      9. *-commutative 98.2%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

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

   5. Taylor expanded in y around 0 60.9%

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

      1. mul-1-neg 60.9%

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

      2. distribute-rgt-neg-in 60.9%

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

   7. Simplified 60.9%

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

   if -1.4e55 < x < -2.29999999999999991e-29 or 5.1000000000000003e206 < x < 3.89999999999999976e251

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 65.2%

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

   if -2.29999999999999991e-29 < x < 6.2e9

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 75.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-29}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6200000000:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+206} \lor \neg \left(x \leq 3.9 \cdot 10^{+251}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+19} \lor \neg \left(y \leq 9.5 \cdot 10^{+69}\right) \land y \leq 5 \cdot 10^{+130}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e+55)
   (* x y)
   (if (or (<= y 2e+19) (and (not (<= y 9.5e+69)) (<= y 5e+130)))
     (* z (- 1.0 x))
     (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e+55) {
		tmp = x * y;
	} else if ((y <= 2e+19) || (!(y <= 9.5e+69) && (y <= 5e+130))) {
		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 <= (-5d+55)) then
        tmp = x * y
    else if ((y <= 2d+19) .or. (.not. (y <= 9.5d+69)) .and. (y <= 5d+130)) 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 <= -5e+55) {
		tmp = x * y;
	} else if ((y <= 2e+19) || (!(y <= 9.5e+69) && (y <= 5e+130))) {
		tmp = z * (1.0 - x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e+55:
		tmp = x * y
	elif (y <= 2e+19) or (not (y <= 9.5e+69) and (y <= 5e+130)):
		tmp = z * (1.0 - x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e+55)
		tmp = Float64(x * y);
	elseif ((y <= 2e+19) || (!(y <= 9.5e+69) && (y <= 5e+130)))
		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 <= -5e+55)
		tmp = x * y;
	elseif ((y <= 2e+19) || (~((y <= 9.5e+69)) && (y <= 5e+130)))
		tmp = z * (1.0 - x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e+55], N[(x * y), $MachinePrecision], If[Or[LessEqual[y, 2e+19], And[N[Not[LessEqual[y, 9.5e+69]], $MachinePrecision], LessEqual[y, 5e+130]]], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000046e55 or 2e19 < y < 9.4999999999999995e69 or 4.9999999999999996e130 < y

   1. Initial program 97.7%

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

   2. Taylor expanded in y around inf 73.5%

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

   if -5.00000000000000046e55 < y < 2e19 or 9.4999999999999995e69 < y < 4.9999999999999996e130

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+19} \lor \neg \left(y \leq 9.5 \cdot 10^{+69}\right) \land y \leq 5 \cdot 10^{+130}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-42} \lor \neg \left(x \leq 9 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.5e-42) (not (<= x 9e-6))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5e-42) || !(x <= 9e-6)) {
		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 <= (-2.5d-42)) .or. (.not. (x <= 9d-6))) 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 <= -2.5e-42) || !(x <= 9e-6)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.5e-42) or not (x <= 9e-6):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.5e-42) || !(x <= 9e-6))
		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 <= -2.5e-42) || ~((x <= 9e-6)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.5e-42], N[Not[LessEqual[x, 9e-6]], $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 < -2.50000000000000001e-42 or 9.00000000000000023e-6 < x

   1. Initial program 98.6%

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

      1. sub-neg 98.6%

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

      2. +-commutative 98.6%

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

      3. distribute-lft1-in 98.6%

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

      4. associate-+r+ 98.6%

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

      5. +-commutative 98.6%

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

      6. *-commutative 98.6%

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

      7. neg-mul-1 98.6%

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

      8. associate-*r* 98.6%

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

      9. *-commutative 98.6%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.7%

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

   if -2.50000000000000001e-42 < x < 9.00000000000000023e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-42} \lor \neg \left(x \leq 9 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.9% 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 98.5%

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

      1. sub-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. distribute-lft1-in 98.5%

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

      4. associate-+r+ 98.5%

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

      5. +-commutative 98.5%

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

      6. *-commutative 98.5%

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

      7. neg-mul-1 98.5%

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

      8. associate-*r* 98.5%

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

      9. *-commutative 98.5%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in y around inf 99.2%

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

4. Final simplification 98.9%

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

Alternative 6: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e-24) (* x y) (if (<= x 3.5e-18) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e-24) {
		tmp = x * y;
	} else if (x <= 3.5e-18) {
		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 <= (-3.2d-24)) then
        tmp = x * y
    else if (x <= 3.5d-18) 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 <= -3.2e-24) {
		tmp = x * y;
	} else if (x <= 3.5e-18) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.2e-24:
		tmp = x * y
	elif x <= 3.5e-18:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e-24)
		tmp = Float64(x * y);
	elseif (x <= 3.5e-18)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.2e-24)
		tmp = x * y;
	elseif (x <= 3.5e-18)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e-24], N[(x * y), $MachinePrecision], If[LessEqual[x, 3.5e-18], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000012e-24 or 3.4999999999999999e-18 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in y around inf 47.7%

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

   if -3.20000000000000012e-24 < x < 3.4999999999999999e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

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 99.2%

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

   1. sub-neg 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-lft1-in 99.2%

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

   4. associate-+r+ 99.2%

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

   5. +-commutative 99.2%

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

   6. *-commutative 99.2%

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

   7. neg-mul-1 99.2%

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

   8. associate-*r* 99.2%

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

   9. *-commutative 99.2%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 36.6% 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. Taylor expanded in x around 0 37.5%

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

3. Final simplification 37.5%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023195 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))