Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

Alternatives: 9

Speedup: 1.1×

• 65.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. distribute-lft-neg-out N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00027:\\ \;\;\;\;\mathsf{fma}\left(z, x - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (* y (- t x))))
  (if (<= y -8.2e+30)
    t_1
    (if (<= y 0.00027) (fma z (- x t) x) (+ x t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -8.2e+30) {
		tmp = t_1;
	} else if (y <= 0.00027) {
		tmp = fma(z, (x - t), x);
	} else {
		tmp = x + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -8.2e+30)
		tmp = t_1;
	elseif (y <= 0.00027)
		tmp = fma(z, Float64(x - t), x);
	else
		tmp = Float64(x + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.2e+30], t$95$1, If[LessEqual[y, 0.00027], N[(z * N[(x - t), $MachinePrecision] + x), $MachinePrecision], N[(x + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2000000000000001e30

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.8%

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

   9. Applied rewrites 45.8%

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

   if -8.2000000000000001e30 < y < 2.7e-4

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, x - t, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 59.5%

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

   if 2.7e-4 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 61.4%

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

   4. Applied rewrites 61.4%

      \[\leadsto x + \color{blue}{y \cdot \left(t - x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00027:\\ \;\;\;\;\mathsf{fma}\left(z, x - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (* y (- t x))))
  (if (<= y -8.2e+30) t_1 (if (<= y 0.00027) (fma z (- x t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -8.2e+30) {
		tmp = t_1;
	} else if (y <= 0.00027) {
		tmp = fma(z, (x - t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -8.2e+30)
		tmp = t_1;
	elseif (y <= 0.00027)
		tmp = fma(z, Float64(x - t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.2e+30], t$95$1, If[LessEqual[y, 0.00027], N[(z * N[(x - t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2000000000000001e30 or 2.7e-4 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.8%

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

   9. Applied rewrites 45.8%

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

   if -8.2000000000000001e30 < y < 2.7e-4

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, x - t, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, x - t, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(z - y, x, x\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fma (- z y) x x)))
  (if (<= x -7.5e-46) t_1 (if (<= x 1.7e+22) (* t (- y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z - y), x, x);
	double tmp;
	if (x <= -7.5e-46) {
		tmp = t_1;
	} else if (x <= 1.7e+22) {
		tmp = t * (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z - y), x, x)
	tmp = 0.0
	if (x <= -7.5e-46)
		tmp = t_1;
	elseif (x <= 1.7e+22)
		tmp = Float64(t * Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[x, -7.5e-46], t$95$1, If[LessEqual[x, 1.7e+22], N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000003e-46 or 1.7e22 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.5%

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

   6. Applied rewrites 55.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 55.5%

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

   8. Applied rewrites 55.5%

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

   if -7.5000000000000003e-46 < x < 1.7e22

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-z, t, x\right)\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-neg.f64 N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 64.9%

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 49.5%

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

   11. Applied rewrites 49.5%

       \[\leadsto \color{blue}{t \cdot \left(y - z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ t_2 := t \cdot \left(y - z\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7.1 \cdot 10^{-78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-290}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (* y (- t x))) (t_2 (* t (- y z))))
  (if (<= y -4.6e+31)
    t_1
    (if (<= y -7.1e-78)
      t_2
      (if (<= y -7.8e-130)
        (fma z x x)
        (if (<= y 9.8e-290)
          (* z (- x t))
          (if (<= y 3.5e-114)
            (fma z x x)
            (if (<= y 8.5e-5) t_2 t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double t_2 = t * (y - z);
	double tmp;
	if (y <= -4.6e+31) {
		tmp = t_1;
	} else if (y <= -7.1e-78) {
		tmp = t_2;
	} else if (y <= -7.8e-130) {
		tmp = fma(z, x, x);
	} else if (y <= 9.8e-290) {
		tmp = z * (x - t);
	} else if (y <= 3.5e-114) {
		tmp = fma(z, x, x);
	} else if (y <= 8.5e-5) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	t_2 = Float64(t * Float64(y - z))
	tmp = 0.0
	if (y <= -4.6e+31)
		tmp = t_1;
	elseif (y <= -7.1e-78)
		tmp = t_2;
	elseif (y <= -7.8e-130)
		tmp = fma(z, x, x);
	elseif (y <= 9.8e-290)
		tmp = Float64(z * Float64(x - t));
	elseif (y <= 3.5e-114)
		tmp = fma(z, x, x);
	elseif (y <= 8.5e-5)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+31], t$95$1, If[LessEqual[y, -7.1e-78], t$95$2, If[LessEqual[y, -7.8e-130], N[(z * x + x), $MachinePrecision], If[LessEqual[y, 9.8e-290], N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-114], N[(z * x + x), $MachinePrecision], If[LessEqual[y, 8.5e-5], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.5999999999999999e31 or 8.5000000000000006e-5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.8%

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

   9. Applied rewrites 45.8%

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

   if -4.5999999999999999e31 < y < -7.1000000000000002e-78 or 3.5e-114 < y < 8.5000000000000006e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-z, t, x\right)\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-neg.f64 N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 64.9%

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 49.5%

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

   11. Applied rewrites 49.5%

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

   if -7.1000000000000002e-78 < y < -7.8000000000000002e-130 or 9.8000000000000002e-290 < y < 3.5e-114

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 37.2%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.2%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. +-commutative N/A

          \[\leadsto z \cdot x + x \]

       5. lower-fma.f64 37.2%

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

   11. Applied rewrites 37.2%

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

   if -7.8000000000000002e-130 < y < 9.8000000000000002e-290

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 44.0%

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

   6. Applied rewrites 44.0%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ t_2 := t \cdot \left(y - z\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7.1 \cdot 10^{-78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (* y (- t x))) (t_2 (* t (- y z))))
  (if (<= y -4.6e+31)
    t_1
    (if (<= y -7.1e-78)
      t_2
      (if (<= y 3.5e-114) (fma z x x) (if (<= y 8.5e-5) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double t_2 = t * (y - z);
	double tmp;
	if (y <= -4.6e+31) {
		tmp = t_1;
	} else if (y <= -7.1e-78) {
		tmp = t_2;
	} else if (y <= 3.5e-114) {
		tmp = fma(z, x, x);
	} else if (y <= 8.5e-5) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	t_2 = Float64(t * Float64(y - z))
	tmp = 0.0
	if (y <= -4.6e+31)
		tmp = t_1;
	elseif (y <= -7.1e-78)
		tmp = t_2;
	elseif (y <= 3.5e-114)
		tmp = fma(z, x, x);
	elseif (y <= 8.5e-5)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+31], t$95$1, If[LessEqual[y, -7.1e-78], t$95$2, If[LessEqual[y, 3.5e-114], N[(z * x + x), $MachinePrecision], If[LessEqual[y, 8.5e-5], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5999999999999999e31 or 8.5000000000000006e-5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.8%

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

   9. Applied rewrites 45.8%

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

   if -4.5999999999999999e31 < y < -7.1000000000000002e-78 or 3.5e-114 < y < 8.5000000000000006e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y, \mathsf{fma}\left(-z, t, x\right)\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lift-neg.f64 N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 64.9%

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 49.5%

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

   11. Applied rewrites 49.5%

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

   if -7.1000000000000002e-78 < y < 3.5e-114

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 37.2%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.2%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. +-commutative N/A

          \[\leadsto z \cdot x + x \]

       5. lower-fma.f64 37.2%

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

   11. Applied rewrites 37.2%

       \[\leadsto \mathsf{fma}\left(z, x, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00027:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (* y (- t x))))
  (if (<= y -8.2e+30) t_1 (if (<= y 0.00027) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -8.2e+30) {
		tmp = t_1;
	} else if (y <= 0.00027) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -8.2e+30)
		tmp = t_1;
	elseif (y <= 0.00027)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.2e+30], t$95$1, If[LessEqual[y, 0.00027], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2000000000000001e30 or 2.7e-4 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.8%

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

   9. Applied rewrites 45.8%

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

   if -8.2000000000000001e30 < y < 2.7e-4

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. distribute-lft-neg-out N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 37.2%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.2%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. +-commutative N/A

          \[\leadsto z \cdot x + x \]

       5. lower-fma.f64 37.2%

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

   11. Applied rewrites 37.2%

       \[\leadsto \mathsf{fma}\left(z, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 37.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return fma(z, x, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. distribute-lft-neg-out N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 55.5%

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

6. Applied rewrites 55.5%

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

7. Taylor expanded in y around 0

   \[\leadsto x + \color{blue}{x \cdot z} \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 37.2%

      \[\leadsto x + x \cdot z \]

9. Applied rewrites 37.2%

   \[\leadsto x + \color{blue}{x \cdot z} \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. lift-*.f64 N/A

       \[\leadsto x + x \cdot z \]

    3. *-commutative N/A

       \[\leadsto x + z \cdot x \]

    4. +-commutative N/A

       \[\leadsto z \cdot x + x \]

    5. lower-fma.f64 37.2%

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

11. Applied rewrites 37.2%

    \[\leadsto \mathsf{fma}\left(z, x, x\right) \]
12. Add Preprocessing

Alternative 9: 21.8% accurate, 2.9× speedup

Math

\[x \cdot z \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (* x z))

C

double code(double x, double y, double z, double t) {
	return x * z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * z
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * z;
}

Python

def code(x, y, z, t):
	return x * z

Julia

function code(x, y, z, t)
	return Float64(x * z)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * z;
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. distribute-lft-neg-out N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 44.0%

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

6. Applied rewrites 44.0%

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

7. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{z} \]
8. Step-by-step derivation

   1. lower-*.f64 21.8%

      \[\leadsto x \cdot z \]

9. Applied rewrites 21.8%

   \[\leadsto x \cdot \color{blue}{z} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64
  (+ x (* (- y z) (- t x))))